An Addendum to Time Dilation
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In 1936 Carl David Anderson (1905 Sep 03 - 1991 Jan 11), Seth Henry Neddermeyer (1907 ___ - 1988 Jan 20), Edward Stevenson (? - ?), and Jabez Curry Street (1906 May 05 - 1989 Nov 07) established a modified version of the Wilson cloud chamber on Pike's Peak, 4300 meters (14,000 feet) above sea level, and recorded the tracks made in the chamber by the debris spewed from collisions between cosmic rays and the nuclei of atoms high in Earth's atmosphere. Analysis of the photographs led Anderson and Neddermeyer to infer the existence of a subnuclear particle carrying about 200 times the mass of the electron. At first physicists believed that the "mu meson" was the particle that Hideki Yukawa (1907 Jan 23 - 1981 Sep 08) had described theoretically as the carrier of the nuclear force, but subsequent study showed that the mu meson interacted only weakly with atomic nuclei and, thus, was not a meson at all: renamed muon, it turned out to be a lepton (like the electron), which provoked Isidor I. Rabi (1898 Jul 29 - 1988 Jan 11) to utter his famous quip, "Who ordered that?" (While it won't tell who ordered it, the Map of Physics should eventually tell us why the muon exists).
In 1939 the team of Rossi, Van Norman, and Hilbery determined the basic properties of the muon, especially its proper half-life. That last datum gave Rossi (Bruno Benedetti Rossi: 1905 Apr 13 - 1993 Nov 23) and David B. Hall (? - ?) the information necessary to carry out an experiment that they conducted and then reported in 1941 (Rossi, B, and Hall, D.B., Physical Review 59, Pg 223 (1941) On Muon Time Dilation).
Rossi and Hall established their equipment near the summit of Mount Washington, about 6300 feet (2000 meters) above sea level. With its attendant photo-multiplier, camera, and other equipment necessary to the recording and counting of the little flashes of light that signified muons decaying, the apparatus consisted primarily of a block of doped plastic with a thick layer of iron suspended over it. The dopant dissolved in the hard, transparent plastic was a chemical that emitted light when struck by energetic electrons of the kind that emerge from decaying muons. That apparatus gave Rossi and Hall the means to count the muons brought to rest in the plastic.
They made the iron thick enough that muons entering it vertically at speeds close to the speed of light would slow down enough that those muons originally flying at 99.4% of lightspeed, give or take a small fraction of one percent, would come to rest inside the plastic block. Those muons that originally moved slower than that speed came to rest inside the iron and decayed there; those originally moving faster passed through both the iron and the plastic and, thus, also did not add to the count.
On Mount Washington that apparatus detected 563 muons decaying every hour in the plastic block. Near sea level that same apparatus, with some of the iron removed to compensate for the extra 2000 meters of air through which the muons had to travel, detected 412 muons decaying every hour, substantially more than the 25 decays per hour that the men calculated from their Mount Washington data on the assumption that the moving muons did not suffer time dilation. That result matches up perfectly with the result that David H. Frisch and James H. Smith got when they carried out almost exactly the same experiment twenty-one years later.
For both teams the half-life of the muon gave them the key datum through which they could interpret their results. If you search muon-related articles on the Internet, as I have done, you will find that an unfortunate number of those articles give the half-life of the muon as 2.2 microseconds. As Wolfgang Pauli (1900 Apr 25 - 1958 Dec 15) once said of a different subject, that's not even wrong. The authors of those articles have confused half-life with lifetime and if that confusion persists, we can look forward to some serious miscalculations.
Exponential decay does not give us a clear marker that we can use to assign a characteristic time to the decay. We have to contrive a purely arbitrary criterion that will enable us to work out the corresponding characteristic times. Physicists have devised two such criteria, one giving us the half-life and the other giving us the lifetime of the unstable particle.
We define half-life as the interval that elapses between the instant when we obtain a number of unstable particles and the instant when half of those particles remain undecayed. For the muon the half-life equals 1.523 microseconds. Of the physicists who study decaying things, the nuclear physicists (those who study the structure of the atom's nucleus) tend to use half-life.
We define lifetime as the interval that elapses between the instant when we obtain a number of unstable particles and the instant when one napierth remain undecayed (one napierth is the fraction obtained by dividing one by the base of the natural or Napierian logarithms: 1/e = 1/2.718281828...= 0.367879441....). For the muon the lifetime equals 2.197 microseconds. Of the physicists who study decaying things, the high-energy physicists (those who study sub-nuclear particles) tend to use lifetime.
Thus, we see that if we have
1000 muons when our clock reads 0.000, then when the clock reads 1.523
microseconds we will have 500 muons, give or take a few, and when the clock
reads 2.197 microseconds we will have 368 muons, give or take a few. Because of
the way in which we define half-life and lifetime, we also know that the
half-life of a particle equals the lifetime of that particle multiplied by the
natural logarithm of two. And that should clear up that confusion.
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