FOCUSING AND DEFLECTING COSMIC-RAY MUONS

Yes, muons can in principle be focused or redirected into a smaller area, because muons are electrically charged particles.

However, cosmic-ray muons are relatively energetic particles, typically with energies of several GeV, so they are difficult to bend appreciably.

1. PASSIVE FOCUSING DOES NOT WORK

You cannot make a simple “muon funnel” from:

• Lead
• Aluminium
• Steel
• Plastic
• Mirrors
• A cone or horn of ordinary material

Muons do not reflect from surfaces like light.

Putting material around the detector generally causes absorption, energy loss and scattering rather than directing muons towards the detector.

2. MAGNETIC FOCUSING

Because a muon has electric charge, its trajectory can be bent by a magnetic field.

The approximate radius of curvature is:

r = p / (0.3 B)

where:

r = radius in metres
p = muon momentum in GeV/c
B = magnetic field in tesla

For a 3 GeV/c muon:

B = 0.1 T     r = 100 m
B = 0.5 T     r = 20 m
B = 1.0 T     r = 10 m
B = 2.0 T     r = 5 m

Therefore even a 1 tesla field produces quite gentle curvature.

3. FIELD STRENGTH IS NOT EVERYTHING

The important quantity is:

B x L

where L is the distance travelled through the magnetic field.

The approximate deflection angle is:

theta = 0.3 B L / p

For example:

B = 1 tesla
L = 0.10 m
p = 3 GeV/c

gives:

theta = 0.3 x 1 x 0.10 / 3

      = 0.01 radians

      = about 0.57 degrees

This sounds very small, but the deflection becomes measurable if the muon subsequently travels a reasonable distance.

4. EXAMPLE

Suppose the muon passes through:

1 T magnetic field
for 10 cm

and then travels:

1 metre

to another detector.

A 3 GeV/c muon would be displaced by approximately:

10 mm

A rough comparison would be:

Muon momentum Displacement after 1 m

0.5 GeV/c about 60 mm 1 GeV/c about 30 mm 2 GeV/c about 15 mm 3 GeV/c about 10 mm 5 GeV/c about 6 mm 10 GeV/c about 3 mm

This ignores the additional small displacement produced while the particle is actually inside the magnet.

5. A PRACTICAL COSMIC-MUON SPECTROMETER

Rather than trying to focus lots of muons onto one detector, a more practical amateur experiment would be to measure their magnetic deflection.

For example:

         COSMIC MUON
              |
              V

       +-------------+
       | DETECTOR A  |
       +-------------+
              |
              |
       +-------------+
       | DETECTOR B  |
       +-------------+
              |
              V
    +-------------------+
    |                   |
    |   MAGNETIC FIELD  |  <--- 0.5 to 1 T
    |                   |
    +-------------------+
              |
              |
              |     curved trajectory
              |        /
              |       /
              |      /
              V     V

          +-------------+
          | DETECTOR C  |
          +-------------+

Detector C could be moved sideways.

You would measure the triple coincidence rate:

A + B + C

at different positions of detector C.

6. WHY USE THREE DETECTORS?

Detectors A and B establish the incoming trajectory.

The magnet then bends the muon.

Detector C measures where the muon goes afterwards.

Without the first two detectors, you would not know whether a muon arrived at detector C because of magnetic deflection or simply because it originally entered at an angle.

7. POSITIVE AND NEGATIVE MUONS

Cosmic rays contain both:

mu+
mu-

They bend in opposite directions in the same magnetic field.

For example:

                     mu+
                      /
                     /
                    /

Incoming ————->

                    \
                     \
                      \
                     mu-

Therefore you could potentially place detectors on BOTH sides of the original trajectory.

This might eventually allow an experiment investigating the ratio of positive to negative atmospheric muons.

8. REVERSING THE MAGNET

Reversing the magnetic field would be particularly useful.

If an apparent displacement really results from magnetic deflection, reversing the field should reverse the displacement.

This provides an excellent experimental control.

9. HOW STRONG A MAGNET?

For a small cosmic-muon experiment:

Below 0.1 T       Probably rather difficult

0.3 - 0.5 T       Potentially useful

About 1 T         Very useful

Above 1 T         Increasingly useful but
                  increasingly difficult and
                  expensive

However, a tiny 1 T magnet isn’t necessarily useful.

For example:

1 T over 1 cm

is much less useful than:

1 T over 10 cm

because the latter gives ten times the integrated magnetic field.

10. HOW BIG IS A 1 TESLA MAGNET?

There isn’t a particular physical size corresponding to 1 tesla.

A small neodymium magnet may have a surface field approaching perhaps 0.5 T or more.

But this does NOT mean that there is a 0.5 T field extending 10 cm away from it.

The field falls rapidly with distance.

For muon experiments you need:

Strong magnetic field

       AND

Reasonably large aperture

       AND

Reasonably long magnetic path

Producing approximately 1 T across a useful gap can therefore require a substantial magnetic assembly.

11. PERMANENT MAGNET APPROACH

One possibility is opposing powerful neodymium magnets with an iron return yoke:

      IRON RETURN YOKE
   +---------------------+
   |                     |
   |   N             S   |
   | +---+         +---+ |
   | |   | ======> |   | |
   | +---+  FIELD  +---+ |
   |        GAP          |
   |                     |
   +---------------------+

The muons pass through the gap.

The iron yoke concentrates the magnetic flux and provides a return path.

Such an arrangement is considerably more useful than simply placing a single neodymium block beside the muon detector.

12. ELECTROMAGNET

A laboratory electromagnet could produce approximately 0.5-1 T across a useful pole gap.

However, it could:

• Weigh tens of kilograms
• Require considerable electrical power
• Generate substantial heat
• Require a heavy iron magnetic circuit

The advantage is that the field can easily be changed or reversed.

13. FOCUSING VERSUS SPECTROMETRY

Actually concentrating cosmic muons from a large collecting area onto a tiny detector is extremely difficult.

For example:

1 square metre collecting area

         |
         V

     magnetic
      funnel

         |
         V

     5 x 5 cm
     detector

would require a very substantial magnetic system.

A much more achievable experiment is:

Do not try to collect more muons.

Instead:

Select their direction
       |
       V
Bend them magnetically
       |
       V
Measure the displacement
       |
       V
Estimate their momentum

14. BASIC PRINCIPLE

The experiment effectively becomes:

   MUON
     |
     V
+----------+
| DIRECTION|
| DETECTORS|
+----------+
     |
     V
+----------+
| MAGNET   |
+----------+
     |
     | deflection depends
     | on momentum
     V
+----------+
| POSITION |
| DETECTOR |
+----------+

Large deflection:

Lower momentum muon

Small deflection:

Higher momentum muon

CONCLUSION

It is physically possible to focus or steer muons because they are charged particles.

Ordinary materials cannot act as a muon funnel.

Magnetic fields can bend them, but atmospheric cosmic-ray muons are energetic enough that strong fields and/or long magnetic paths are required.

For an amateur experiment using Cosmic Watch or PicoMuon detectors, attempting to concentrate a large area of cosmic muons onto a small detector would be difficult.

A much more practical and interesting experiment would be a magnetic cosmic-muon spectrometer using:

2 detectors to establish incoming direction

0.5-1 T magnetic field

5-10 cm or more magnetic path

50-100 cm flight distance

movable third detector

Such an apparatus could potentially demonstrate magnetic deflection of cosmic muons and investigate their momentum distribution and, with a more sophisticated arrangement, the opposite deflection of positive and negative muons.