Question 14.1: Find the length of the tracks produced in a bubble chamber b...
Find the length of the tracks produced in a bubble chamber by a particle traveling with a speed equal to 0.96 c that decays by the weak interaction in 10^{−10} s. What would the length of the track be if the particle were to decay by the electromagnetic interaction in 10^{−16} s, or the strong interaction in 10^{−24} s?
Using Eq. (12.26) of Chapter 12, the lifetime of the particle in the laboratory frame of reference would be
\Delta t_{M}=\frac{\Delta t_{R}}{\sqrt{1-u^{2}/c^{2}} }. (12.26)
\Delta t=\frac{10^{-10} \ s}{\sqrt{1-(0.96)^{2}} }=3.57\times 10^{-10} \ s.Hence, the length of the track in the bubble chamber would be
\Delta x = 0.96 c × 3.57 × 10^{−10} \ s.Using the value for the velocity of light given in Appendix A, we obtain
| Appendix A | |||
| Constants and conversion factors | |||
| Constants | |||
| Speed of light | c | 2.99792458 × 10^{8} m/s | |
| Charge of electron | e | 1.6021773 × 10^{−19} C | |
| Plank’s constant | h | 6.626076 × 10^{−34} J s | |
| 4.135670 × 10^{−15} eV s | |||
| \hbar=h/2π | 1.054573 × 10^{−34} J s | ||
| 6.582122 × 10^{−16} eV s | |||
| hc | 1239.8424 eV nm | ||
| 1239.8424 MeV fm | |||
| Hydrogen ionization energy | 13.605698 eV | ||
| Rydberg constant | 1.0972 × 10^{5} cm^{−1} | ||
| Bohr radius | a_{0} = (4π \epsilon _{0})/(me²) | 5.2917725 × 10^{−11} m | |
| Bohr magneton | μ_{B} | 9.2740154 × 10^{−24} J/T | |
| 5.7883826 × 10^{−5} eV/T | |||
| Nuclear magneton | μ_{N} | 5.0507865 × 10^{−27} J/T | |
| 3.1524517 × 10^{−8} eV/T | |||
| Fine structure constant | α = e^{2}/(4π\epsilon _{0} c \hbar) | 1/137.035989 | |
| e^{2}/4π\epsilon _{0} | 1.439965 eV nm | ||
| Boltzmann constant | k | 1.38066 × 10^{−23} J/K | |
| 8.6174 × 10^{−5} eV/K | |||
| Avogadro’s constant | N_{A} | 6.022137 × 10^{23} mole | |
| Stefan-Boltzmann constant | σ | 5.6705 × 10^{−8} W/m² K^{4} | |
| Particle masses | |||
| kg | u | MeV/c² | |
| Electron | 9.1093897 × 10^{−31} | 5.485798 × 10^{−4} | 0.5109991 |
| Proton | 1.6726231 × 10^{−27} | 1.00727647 | 938.2723 |
| Neutron | 1.674955 × 10^{−27} | 1.008664924 | 939.5656 |
| Deuteron | 3.343586 × 10^{−27} | 2.013553 | 1875.6134 |
| Conversion factors | |||
| 1 eV | 1.6021773 × 10^{−19} J | ||
| 1 u | 931.4943 MeV/c² | ||
| 1.6605402 × 10^{−27} kg | |||
| 1 atomic unit | 27.2114 eV | ||
The length of the track is thus about 10.3 cm long and could be easily observed.
Using the same approach, the length of the track left by a particle that decayed by the electromagnetic interaction would be 0.1027× 10^{−6} m and the length of the track of a particle decaying by the strong interaction would be 0.1027× 10^{−14} m = 1.027 fm. While it might be possible to observe the track of a rapidly moving particle that decays electromagnetically, the length of the path of a particle that decays by the strong interaction would be about equal to the radius of an atomic nucleus and would not be observable.
