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184 chapter 840fitNumber20dataN(t)0040 80 120t [ns]Figure 8.4 A reproduction of the experimental measurement in [Stez 73] of the number ofdecays of a π meson as a function of time. Measurements are made during time intervals of10-ns length. Each “event” corresponds to a single decay.Here τ =1/λ is the lifetime of the particle, with λ the rate parameter. The actualdecay rate is given by the second equation in (8.47). If the number of decays ∆N isvery small compared to the number of particles N, and if we look at vanishinglysmall time intervals, then the difference equation (8.47) becomes the differentialequationdN(t)dt≃−λN(t)= 1 N(t). (8.48)τThis differential equation has an exponential solution for the number as well as forthe decay rate:N(t)=N 0 e −t/τ ,dN(t)dt= − N 0τ e−t/τ = dN(0) e −t/τ . (8.49)dtEquation (8.49) is the theoretical formula we wish to “fit” to the data in Figure 8.4.The output of such a fit is a “best value” for the lifetime τ.8.7 Least-Squares Fitting (Method)Books have been written and careers have been spent discussing what is meant bya “good fit” to experimental data. We cannot do justice to the subject here and refer−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 184