The work described here is based on a series of articles (MacLeod et al 1976, MacLeod 1976, 1980) in which a laboratory radioactive source was used as a source of random events, which enabled an investigation of the Poisson distribution to be made, and was of considerable benefit in giving undergraduate students an understanding of counting statistics

Mulligan, F.J.

English

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Eur. J. Phyr. 8 (1987) 53-57. Prlnted in the UK Letters and comments Experimental statistics for undergraduates-revisited 1. Introduction The  work  described  here is based on a  series  of  articles (MacLeod et a1 1976, MacLeod 1976, 1980) in which a laboratory radioactive source was used as a source of random events, which enabled an investigation of the  Poisson  distribution  to be made,  and  was of considerable benefit in giving undergraduate students an  understanding of counting  statistics.  The  second article in this series (MacLeod 1976) gives details of the  circuitry  used  for counting, while the  third (MacLeod 1980) describes  the  circuitry  used  to  obtain a graphic display of the data on a television screen. The second article is an interesting and informative exercise in digital electronics. and the third article is useful in understanding  video  display  circuitry. However, if one is interested  only in the  statistics of the experiment,  both of these  exercises  are unnecessary. The  present  work  shows  that  the  counting  circuitry (34 of the  74  series  integrated  circuits  (MacLeod 1976)) may be replaced by a pair of NAND gates and a 6522 versatile interface adapter (VIA). The display circuitry is already  available in almost  every  popu- lar  make of microcomputer on the  market  oday. Furthermore, many of these micros have a 6522 VIA available  on  board for the user as  an  interface  de- vice, so that  the  only  circuitry  external  tohe microcomputer  for  the  purpose of counting is a  pair of NAND gates.  MacLeod (1980) considers  the possibility of using a microcomputer to perform the functions outlined in his article, but dismisses the idea on the basis  that no significant  reduction in specialised hardware  would  result.  This  work  shows  that with the aid of a  6522  VIA,  a very  significant reduction in specialised  hardware is obtained. while the  features of the dynamic display, outlined by MacLeod. are still retained. 2. Apparatus A radioactive  source, amplifier and pulse squarer similar to  those  described  by  MacLeod et a1 (1 976)  are used to provide a continuous stream of TTL pulses occurring  at  random  intervals. One then  eeds  to count repeatedly the number of pulses occurring in a given fixed period. with some  method of recording each  result.  Finally  a  histogram  plot  of  the  frequency of a given count  against  count  number,  together with a method  of  calculating  the  mean,  variance  and x 2  fit are required. The central element in this work is the 6522 VIA, 5 3  which is a sophisticated integrated circuit used as a peripheral interface to many microprocessors. I t  has 16 internal registers. but we will be concerned with only three of these, labelled as T , ,   T z  and CONTROL register. T I  and T 2  are both 16 bit registers and they may be used as  either  timers or counters  depending on the value in the CONTROL register. T ,  is used here to provide  a fixed time  interval. while T 2  is used to  count all the  pulses  that  occur  during  this  interval.  When T ,  is used as a timer in the one-shot mode (cf SY6522 data  sheet,  Synertek  Inc.),  an  output pulse appears on pin P B ~  of the  6522.  the length of which is proportional to the value originally loaded into the T ,  timer. This pulse is then used to gate the random pulses via the two NAND gates,  from  the  radioactive  source to pin P B ~  of the 6522. When T2 is initialised to act as a coun- ter. all pulses fed to pin P B ~  are  recorded  up  to  a maximum of the clocking rate of the 6522 ( I  MHz). Figure 1 shows the simplicity of the interface to the 6522,  where  gate G, is used to  invert the output pulse on P B ~ .  With a clocking rate of 1 MHz the length of the pulse from the T ,  timer may take any value in the range 1-65 535 p ;  typically times in the range 1-65 ms are used as a suitable counting period. The results in this  article were obtained using a  6522 onboard  a BBC microcomputer  and  the  data were plotted using the  graphics  available.  After  ach counting period, the value of the T 2  counter is read and  the  counter is then  restarted.  During  the next counting  period,  the  data  from  the  previous  sample  are Figure I Interface of random pulses  to microcomputer. The random pulses are either gated through G, to P B ~  for t h e  Poisson distribution or fed directly  to C B I  for the exponential curve. (In the  present  work the 6522 VIA is onboard a BBC microcomputer and all the  required connections are available through the user port). r L >".F PBI pulse 39 C A 2  J l 0143-0807/87/010053 +05 $02.50 G 1987 IOP Publishing Ltd & the European Physical Society 54 Letters and comments accumulated  and  the  display is updated;  this  requires  a maximum of 0.9 ms  when  performed in machine  code. Thus, using a  sampling  period  greater  than 1 ms allows the  counting  and  plotting  to  be  performed simultaneously,  and  almost all the  integration  period is spent in the  counting  mode.  When  the  required number of samples  has been obtained  the  processor is dedicated to calculating the mean, variance and x' fit for  the data. The advantages of the system are that the appara- tus for the  statistics  experiment  may be completely assembled in a matter of minutes, rather than going through  MacLeod's  more  complicated  procedure.  The counting  interval.  the  total  number of samples  and  the histogram  scaling  may all be set in software;  software allows far more flexibility than can ever by achieved with a set of switches. The counting rate is 10 times faster than in the earlier work, allowing data to be accumulated  more  quickly.  Using  a  suite of programs,  the  student  can  be  taken  step by step through  a  learning  sequence,  starting with a single- channel  counter  and plotting the  results  by  hand,  up  to a full multichannel  system with theoretical  curves calculated  and  displayed  along with experimental  data for visual comparison. If a high level language. e.g. B A S I C .  is used for processing and plotting the results the student  may  readily  change  the  program  and may well learn  even  more  from the experiment. A print-out of the  results  may be obtained, which  allows comparison of many sets of results, and serves as a permanent  record for the  student's  laboratory notebook. 2.1 .  Exponential curue MacLeod (1980) shows  how his circuitry  was  used  to demonstrate the exponential curve. by measuring the time between  consecutive  pulses  and  plotting  ahistogram of the frequency of occurrence of a given time against  time.  This useful demonstration  may  also be obtained with the present system by applying the TTL random  pulses  to pin C B I  of  the 6 5 2 2  and using T ,  as  a  timer to measure  the  time  between pulses. A type of direct memory access per 500 events, to ensure minimum distortion to the exponential curve, is used by MacLeod;  the simplicity of the  interface  used  here would be lost if such a scheme were attempted. and thus  an  interrupt  system is used  to  register  the  arrival of each event. This adds a small delay ( 7 p )  to each time and  may  cause  some  distortion of the  exponential for pulses which occur very close together. However, because of the ten-fold increase in clock  speed,  to- gether with relatively low count rates (easily adjusted by increasing the distance between source and detec- tor),  quite  acceptable  results  have been achieved as shown in figure 3. 3. Software The  program  to  access  the 6522 VIA and  plot  he histogram of the  results  was  originally  written in BASIC. Each  point  in  the  display  required 20 ms  to  be accumulated  and  plotted, with the  result  that  an integration of 10 000 samples, each of 1 ms duration. required I O  S for the actual counting and 190 S for plotting and  processing.  Nevertheless  a 10 000 sample integration  could  be  obtained in under 4 min, which is a very short time in terms of a  student  laboratory practical  session.  Writing  the  programme in B A S I C  has the added advantage that the lower-level student may readily  make  alterations  to  the  program. The  program  was  rewritten in 6502 code  and each point then required only 0.9 ms to process and plot. Choosing a sampling period of 1 ms or greater allows  the  integration  and  plotting  to  occur  simul- taneously, and almost 100% of the integration time is spent in actual counting. Since the random pulses are gated into the T 2  counter by the T ,  pulse, sam- pling periods as  short  as 1 p may be chosen.  but again  much of the  integration time is spent in processing and plotting rather than in counting. The software to calculate the mean, variance and x' fit is Figure 2 Poisson distributions redrawn from print-out. Bars show  experimental  values. Crosses show theoretical values for a spectrum with the same mean  value. The four figures were obtained with the  following count rates: (a) 860 S". (b)  240 S"? (c) 130 S". ( d )  60 S". r l ' l  t fbl Ii: l 11, .  Id1 Letters and comments 55 Geiger counter pulses are very long ( -2OOps) .  which means that the pulses do not occur at truly random times. However, the effect  of the long dead time is reduced if low count rates are used. Figures 2(a)-(d) show the improvement in x' fit as the count rate was decreased from 800 S" to 6 0  S " .  The figures show that  quite  reasonable  results  may be obtained  at  these count  rates  and  that  the  experiment is readily  available to  almost all physics  laboratories,  without  the need for specialised  hardware.  Figure 3 shows the results of feeding the random pulses to pin C B I  of the 6522  and counting  the  time  between pulses: again  the  results compare  favourably with the  earlier  work. No of samples = 10000.00 Mu = 900.00 Maxnum = 44.00 Chl  squared = 4 2 . 0 1  Figure 3 Exponential  distribution  (time  between pulses) redrawn  from  print-out.  Bars  show  experimental  values  and crosses  show best-fit  exponential.  Maxnum is the  maximum S coordinate  used in the x 2  fit. Mu is the  calculated  number of counts per  second  based on the fit. still written in BASIC, as is the x 2  fit for  the  exponential curve. It is, however, essential to use 6502 code to access  the  6522 in the  case of the  exponential  curve, in order to minimise the delay between the arrival of a pulse  and  the T I  timer being read. 4. Results The results in this work were obtained using a Sr(90) 5 pCi p source  and  asimple  laboratory  Geiger counter. As MacLeod  (1976)  has  pointed  out,  the The entropy of a damped harmonic oscillator This letter arises out of the  Ginsburg-Landau  theory of critical  points  (Shang-Keng Ma  1976), in which fluctuations of the  order  parameter  are  Fourier- analysed,  and  each  term (in a quadratic  approxi- mation)  is  treated as contributing  to  the  partition function as if it were a harmonic oscillator. Usually, however,  the  nergy  attributed  to  the  fluctuation is purely  potential, i.e. proportional  to a:, where is the  amplitude of a component with wavenumber k, and with no term in fi:. T o  use  an  electrical analogy  which will prove helpful later,  the  system is more like an RC circuit  than a resonant LC circuit; or like a very highly overdamped harmonic oscillator. What is the entropy of such a system? On the  answer  to  this  question  depends  the  contribution  of fluctuations  to  the specific heat  anomaly.  For so 5. Conclusion The use  of microcomputers  for  monitoring  experi- ments and collecting data is a commonplace occur- rence in physics  laboratories  today. In this work it is the power and elegance of the 6 5 2 2  interface chip which allows the apparatus to be simplified so much. The  critical  part  of  actually  performing  the  counting is performed  by  the  hardware of the  chip, while the less critical  parts of display  are  performed in software using the  already  available video circuitry. References MacLeod A M 1976 Am. J .  PhJqs. 44 177-80 ~ 1980 E u ~ .  J . PhJ's. 1 88-97 ~ 1985 EuT.J .  P ~ J ' s .  6 65-71 MacLeod A M. Ledingham K W D and  Morton W T 1976 Am J .  Phys. 44 172-6 F J Mulligan St  Patrick's  College,  Maynooth,  Co.  Kildare,  Ireland simple a question the discussions in the literature are surprisingly  complicated,  and  the  treatment given below is an improvement, if only because the means are  commensurate with the  task. An account of earlier work may be recovered from the references cited by Lonzarich (1 986). To derive an expression  for  the  entropy  (which  turns out to involve fewer tedious details relating to con- vergence of integrals than does the free energy), we must include the damping mechanism explicitly - we need to find how  the  entropy of a  complete  system is changed  when it is  linearly  coupled  to  an  oscillator in such  a  way  that  the  oscillator is exponentially  damped. The  circuit of  igure 1 is such  a system, being a simplified version of an  arrangement  analysed else- where  (Pippard  1983). An infinitely long loss-free transmission line, of characteristic  impedance Z O ,  behaves like a  pure  resistance ZO in the  circuit,  and  by varying ZO the  circuit  may  be lightly damped  or 

Experimental statistics for undergraduates - revisited

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Experimental statistics for undergraduates - revisited

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