# Cosmic Ray Muons and Einstein's Special Relativity

The MU Physics Department does not claim to have invented these labs. The exercise below has been done many times in many places, with many variations. Two references are given at the bottom of the page. We believe that each performance of the lab will be unique; in each nature will reveal a slightly different face to the observer.

Go to a workshop on Einstein's Special Relativity

In this demonstration exercise, we detect and identify cosmic ray muons. Classical analysis of their lifetime and the rate at which they arrive from the top of the atmosphere leads to a contradiction. The contradiction is removed if relativistic time dilation is applied to the rapidly moving muons. The experiment uses a 7 gallon jug of water as the muon sensor, an inexpensive photomultiplier as the detector of light flashes caused by the muons in the water, and an expensive multichannel scalar to record the decay of those muons which are captured in the water jug. The purchase of the photomultipliers and the multichannel scalar was funded by the National Science Foundation's Division of Undergraduate Education through grant DUE-96-50912.

The basic experimental set up is shown in the figure. When the signal from the preamplifier is sent to an oscilloscope, pulses are displayed on the oscilloscope. The rate of occurrence of the pulses increases when a small radioactive source is placed near the jug. We associate the light flashes detected by the photomultiplier with particles passing through the water, ionizing atoms as they go.

In operation, the pulses travel first to the trigger input of the multichannel scalar. If a pulse is above the trigger threshold (set by the operator), a timer starts within the scalar. The timer runs for a preset period; 40 microseconds being typical in this experiment. Such an initial pulse is ignored by the discriminator, although it is also applied to the discriminator input.

If a second pulse arrives during that 40 microsecond window, it is ignored by the trigger circuit, but detected at the discriminator input. If it is above the discriminator threshold (set by the operator), the number stored at a particular memory address is increased by 1. That address is determined by the time at which the second pulse arrived. After a day or two of taking data, the memory will display how many secondary pulses arrived at each time following an initial pulse.

The intent is to detect first the light flash that occurs when a muon arrives in the jug, and then to detect the flash Associated with the muon decay. Most muons do not decay in the jug, but enough do for us to make a measurement. Accelerator experiments show that the muons decay with a characteristic time of about 2 microseconds. The decay is exponential so that if a population of muons is observed at any particular time, we know that 2 microseconds later only about 1/3 of that population will remain.

The probability of decay for a muon observed at t=0 is exponential with the same characteristic time as the population. Thus we can expect 2/3 of the decays to occur within about 2 microseconds of the initial pulse. This result is independent of the previous history of the muon. The number of decays, plotted versus the time of the decay, forms an exponential curve, with characteristic time equal to the characteristic lifetime of the muon. This is the graph that the multichannel scalar creates for us.

The figures at the right shows results for the same set up as the figure above it, with one change: 8 lead bricks (20x10x5cm) were placed above the jug, making a stack 25cm high. In both measurements the count stopped after 30,000 initial pulses had been detected (a period of about 2 days). When the bricks were added, the total number of secondary counts (presumed to be decaying muon signatures) increased from 5568 to 6396.

This result is consistent with idea that most muons are too energetic(1) to be stopped by the water, and so will not stay in the jug long enough to decay and be detected. The lead bricks are supposed to slow down some of those fast muons, make them "stoppable," in the water, and create more observed muon decays. The importance of the argument is that it is evidence for the muons arriving from above, and not from the earth.

If the results shown above are taken as typical, then they do indicate that the source of muons is from above. (They remain atypical at present, as we continue to optimize our choice of electronic thresholds.) If the muons come from above, the obvious next question is how far above. Since they live only a few microseconds, the cannot come directly from stars or other distant objects.

The proposed model is that they are created at the top of the atmosphere by collisions of cosmic ray "primary" particles. If those particles are, for example, protons, then they live long enough to make the trip from distant parts of the universe. If the muons are formed at the top of the atmosphere, then they must travel about 10 kilometers down in order to reach our detector. The question of whether they live long enough to get here still arises.

If they travel at the speed of light, it will take a muon about 30 microseconds to reach earth. This is 15 times the muon characteristic decay time. This means that only about 1 muon survives at the earth surface for every 10 million that start out at the top of the atmosphere.

The cross sectional area of our detector is about 0.1 square meter. For every muon flash in our detector, there should be about 10 million such flashes above the detector. We can estimate how bright that light should be. Accelerator experiments show that one muon decay releases an electron with a maximum of 50MeV. We will assume that 1 MeV of that energy is emitted as light. (The energetic electron ionizes water molecules, and light is emitted when the ionized electron is recaptured.)

We see about one flash every 30 seconds, and would expect to see about 1 flash every 1 second for a detector of 3 square meter cross sectional area. That corresponds to light energy of 1 MeV per second, which is about 10-13 Joules per second, or 10-13 watts.

Thus in a column above us, 3 meters square (and 10,000 meters high) we would expect to see 10 million times that much light, from all the muons that decayed before reaching us. Each 3 square meter patch above us should radiate about 1 microwatt of light.

The planet Venus sends about 1 microwatt of light energy to a square meter of earth. This calculation indicates that we should be seeing about "one Venus worth" of light from each 3 square meters of sky above us. The sky should glow enough to make Venus difficult to see.

It does not.

The resolution of this contradiction is found in the special theory of relativity. When we observe a moving object which carries a clock, we see the clock running slow. If we could watch a muon coming down from the top of the atmosphere we would see it's "clock run slow." Two thirds of those muons will decay in about 2 microseconds, as measured on the muon clock.

Since for us that clock runs slow, we observe many more muons surviving at the surface of the earth than we would have expected. Our estimate of one in 10 million being able to reach the earth is wrong, because we neglected the time dilation that is predicted by the theory of relativity.

In this way, the muon observations disagree with classical mechanics and agree with Einstein's special theory of relativity.

Students or teachers who want to work with real muon detector data may contact Dr. Dooley at the Physics Department, Millersville University. Jwdooley@aol.com.

(1)A.O. Weissenberg, Muons, John Wiley & Sons (1967), p316.

The best web reference for muon physics basics is from the California Institute of Technology

http://tweedledee.wonderland.caltech.edu/~derose/labs/exp15.html

It has the most complete discussion of the behavior of cosmic ray muons and an effective list of references.

A good list of references is also available at