Brett Stempel

David Levy

Dan Olson

Knock-On Electrons

            Moving at speeds very near that of light, particles referred to as cosmic rays arrive in the earth’s vicinity with enormous energy.  In some cases, protons arrive with about the same energy as a baseball thrown by a Major League pitcher.  Current research devoted to the study of cosmic rays hopes to explain the origin and even locate the source of these particles.

            When a cosmic ray slams into the earth’s atmosphere, its interaction produces a shower of particles.  The more energetic the cosmic ray, the larger the shower is.  One way to study cosmic rays is to look for coincidences among laterally separated detectors.  The greater the number of detectors and the larger their separation, the larger is the cosmic ray shower which can be monitored.  In the spring of 2001, students from Mount Michael began an experiment in which coincidences recorded by a pair of detectors were tracked as a function of the distance between the detectors. (Figure 1)  The detectors consist of scintillator panels 60 cm by 60 cm.  A photomultiplier tube (PMT) is mounted in the center of the panel.  Signals from the PMTs pass through a discriminator and a logic unit to a scaler where the counts are manually recorded..  Beginning at a distance of 25 m, the coincidences were recorded at one meter intervals as the distance between the two detectors was decreased until the detectors were side by side. A graph of the coincidences versus the distance shows that the number of counts goes down and then levels off at a distance of about 5 meters. (Figure 2)  Coincidences beyond 5 m are mostly attributed to “accidentals,” two unrelated shower particles passing through the detectors at the same time.  Dr. James Cronin of the University of Chicago suggested a plot of the log of the coincidences versus distance would be instructive.  Indeed the graph of the log of coincidences versus distance shows a change in slope at about one meter. (Figure 3)  To ensure that the change in slope was reproducable, the experiment was repeated in the spring of 2002, but this time with two pairs of detectors.  The distance between the detector pairs was increased in 25 cm intervals beginning side by side and extending to a distance of 5 meters.  The results from the experiment are consistent with the previous experiment..  Just as in the log graph from the 2001 data, there is a change in slope at about one meter. (Figure 4)  What accounts for the greater number of coincidences inside one meter?

            In a private communication, Dr James Cronin suggested that the effect is due to “knock-on electrons.”  Basically, a "knock-on electron" is the result of a collision of an ionizing particle, most frequently a muon, with an electron somewhere in Earth's atmosphere.  The muon is moving at very high speeds, close to the speed of light.  As it is traveling, it knocks other particles like electrons out of its path.  In some cases, the muon has so much energy that the electron it "knocks" out of its path also becomes an ionizing particle.  These "knock-on" electrons are detected by the scintillator panels.

            On the basis of conservation of energy and conservation of momentum (including relativistic effects because of the high speeds), the maximum kinetic energy of a "knock-on electrons" can be determined.  For a particle of mass M with relativistic energy E and relativistic momentum p, the maximum kinetic energy imparted to the electron is given as:

                                                T_max = 2m_ep²/[m_e² + M² + 2m_eE/c²].                                          (1)

Define g = (1 - v² /c²)^½.  Then the expression for the maximum kinetic energy can be re-written:

                                         T_max = 2m_e(g² - 1)/[1 + 2g m_e/M + m_e²/M²].                                      (2)

For highly relativistic particles where E is approximately pc, equation (2) can be written as:

                                                T_max = E²/[E + 11]                                                                       (3)

where E is given in GeV.  The average energy of a muon produced in a cosmic ray shower which reaches Earth’s surface is about 4 GeV.  That implies that for most “knock on” electrons, they will have an energy of about 1.07 GeV, or about one-fourth the energy of the muon that caused the electron to become ionizing.

            Once again, Dr. James Cronin suggested an experiment in which two pairs of detectors were connected side by side and two pairs of detectors were connected diagonally.  When the detectors are side by side, one expects to see a larger number of coincidences than when the detectors are connected diagonally. (Figure 5)  An incoming muon from overhead would produce a spray of knock on electrons that would have a circular footprint.  Such a footprint is more likely to trigger both detectors when they are side by side than when they are placed on the diagonal.  The results of the experiment confirm this expectation.  Approximately twice as many counts were seen for side by side detectors than for diagonally connected detectors.  (Figure 6)

            While studying the graphs, it also appears that "knock-on electrons" have a limited lateral range.  This range appears to be at about one meter, when the "knock-on electrons" influence on the results steadily decreases.  After these experiments, a couple conclusions can be reached.  The first is that "knock-on electrons" have an effect on the results of cosmic rays, except the effect is increased at smaller distances.  Additionally, side by side detectors see more counts than diagonal detectors because of the limited range.

Figure 1

Figure 1: Determining coincidences for detectors separated horizontally.

Figure 2

Figure 2: Coincidence rate for two detectors as a function of distance between the detectors.  The coincidence rate reported is an average of six trials at each distance.

Figure 3

Figure 3: A graph of the log of the coincidence rate as function of detector separation distance.  There is a change in the slope at a detector separation of about one meter.  Data from the spring of 2001.

Figure 4

Figure 4: Graphs of the log of the coincidence rate for two detector pairs as a function of detector separation distance.  Again there is a change in the slope at a detector separation of about one meter.  Data from the spring of 2002.

Figure 5

Figure 5: Detectors connected side by side and detectors connected on the diagonal.  For convenience detectors were labeled alphabetically.

Figure 6

Figure 6. Frequency versus count rate.  The dark green represents the count rate for detectors connected diagonally.  The blue represents the count rate for detectors connected side by side.  On the diagonal detectors E and F gave an average count rate of 77.9 coincidences in 30 minutes with a standard deviation of 18.9.  On the diagonal detectors G and H gave an average count rate of 84.7 coincidences in thirty minutes with a standard deviation of 7.2.  Side by side detectors E and F gave an average count rate of 201 coincidences in 30 minutes with a standard deviation of 58.9.  On the diagonal detectors G and H gave an average count rate of 166.7 coincidences in thirty minutes with a standard deviation of 37.2.