|
in a Bubble Chamber |
The data for this experiment is in the form of a bubble chamber
photograph which shows bubble tracks made by elementary particles as they
traverse liquid hydrogen. In the experiment under study, a beam of low-energy
negative pions (p- beam) hits a hydrogen
(p for proton) target inside the bubble chamber. The bubble
chamber is essentially a container with liquid hydrogen normally kept just
below its boiling point (T=20 K). As the pions enter the detector
a piston slightly decompresses the liquid so it becomes ``super-critical''
and starts boiling, and
bubbles form, first at the ionization trails left by the charged
particles traversing the liquid.
![]() Figure 1: Photograph of the interaction between a high-energy p--meson from the Berkeley Bevatron accelerator and a proton in a liquid hydrogen bubble chamber, which produces two neutral particles L0 and K0, which are short-lived and decay into charged particles a bit further. |
Figure 2: Illustration of the interaction, and identification of bubble
trails and variables
to be measured in the photograph in Figure
3.
The reaction shown in Figure 1 shows the production of a pair of neutral particles (that do not leave a ionized trail in their wake), which after a short while decay into pairs of charged particles:
where the neutrals decay as follows:
In this experiment, we assume the masses of the proton (mp = 938.3 MeV/c2) and the pions (mp + = mp- = 139.4 MeV/c2) to be known precisely, and we shall determine the masses of the L0 and the K0, also in these mass energy units.
Note, that the above is strictly true only if all momenta are perfectly in the plane of the photograph; in actual experiments stereo photographs of the interaction are taken to be able to reconstruct the interaction in all three dimensions. The interaction in this photograph was specially selected for its planarity.
In the reproduced photograph the actual radius of curvature R of the track in the bubble chamber is multiplied by the magnification factor m, r = mR. For the reproduction in Figure 3 m = height of photograph (in mm) divided by 173 mm.
The momentum p of the particles is proportional to their radius of curvature R in the chamber. To derive this relationship for relativistic particles we begin with Newton's law in the form:
Here the momentum (p) is the relativistic momentum mvg, where the relativistic g-factor is defined in the usual way
g = [Ö(1-v2/c 2)]-1.
Thus, because the speed v is constant:
where r is the unit vector in the radial direction. Division by v on both sides of the last equality finally yields:
identical to the non-relativistic result! In atomic units we find:
p c (in eV) = c R B , thus p (in MeV/c) = 2.998×108R B ×10-6 = 300 R(in m) B(in T) | (1) |
p+sinq+ = p-sinq- | (2) |
p0 = p+cosq+ + p-cosq- | (3) |
E0 = E+ +
E-,
where E+ = Ö(p+ 2c2 + m+2c4) , and E- = Ö (p-2c2 + m-2c4) |
|
m0c2 = Ö(E02 - p02c2) |
K0 decay.
Bonus question:
(25% extra credit)
Calculate the momenta for both neutral particles, and hence find their
lifetimes, both in the laboratory, and in their own rest-frames. Compare
the latter with the accepted values [3].
[2] “The Particle Adventure”, http:// pdg.lbl.gov/cpep/adventure.html
[3] Review of Particle Physics, by the Particle Data Group,
European
Physical Journal C3 (1998) 1 - 794
(previous edition: Physical
Review
D54 (1996) 1 - 720) (available on WWW: http://pdg.lbl.gov)
[4] Kenneth Krane: Modern Physics, 2nd ed. ; John Wiley &
Sons, New York 1996
Figure 3 (Next Page): Photograph of the interaction between a
high-energy p--meson from the Berkeley
Bevatron accelerator and a proton in a liquid hydrogen bubble chamber.
The interaction produces two neutral particles L0
and K0, which are short-lived and decay into charged
particles. The photo covers an area (H•W) of 173 mm • 138 mm of the bubble
chamber. In this enlargement the magnification factor g = (height
(in mm) of the photograph)/(173 mm).