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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 10−1010^{−10} s. What would the length of the track be if the particle were to decay by the electromagnetic interaction in 10−1610^{−16} s, or the strong interaction in 10−2410^{−24} s?

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Using Eq. (12.26) of Chapter 12, the lifetime of the particle in the laboratory frame of reference would be

ΔtM=ΔtR1−u2/c2.\Delta t_{M}=\frac{\Delta t_{R}}{\sqrt{1-u^{2}/c^{2}} }.               (12.26)

Δt=10−10 s1−(0.96)2=3.57×10−10  s.\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

Δx=0.96c×3.57×10−10  s.\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 2.99792458×108m/s2.99792458 × 10^{8} m/s
Charge of electron e 1.6021773×10−191.6021773 × 10^{−19} C
Plank’s constant h 6.626076×10−346.626076 × 10^{−34} J s
4.135670×10−154.135670 × 10^{−15}  eV s
ℏ=h/2π \hbar=h/2π 1.054573×10−341.054573 × 10^{−34}  J s
6.582122×10−166.582122 × 10^{−16}  eV s
hc 1239.8424 eV nm
1239.8424 MeV fm
Hydrogen ionization energy 13.605698 eV
Rydberg constant 1.0972×105cm−11.0972 × 10^{5} cm^{−1}
Bohr radius  a0=(4πϵ0)/(me²)a_{0} = (4π \epsilon _{0})/(me²) 5.2917725×10−115.2917725 × 10^{−11} m
Bohr magneton μBμ_{B} 9.2740154×10−249.2740154 × 10^{−24}  J/T
5.7883826×10−55.7883826 × 10^{−5}  eV/T
Nuclear magneton μNμ_{N} 5.0507865×10−275.0507865 × 10^{−27}  J/T
3.1524517×10−83.1524517 × 10^{−8}  eV/T
Fine structure constant α=e2/(4πϵ0c ℏ)α = e^{2}/(4π\epsilon _{0} c  \hbar) 1/137.035989
e2/4πϵ0e^{2}/4π\epsilon _{0} 1.439965 eV nm
Boltzmann constant k 1.38066×10−231.38066 × 10^{−23}  J/K
8.6174×10−58.6174 × 10^{−5}  eV/K
Avogadro’s constant NAN_{A} 6.022137×10236.022137 × 10^{23}  mole
Stefan-Boltzmann constant σ 5.6705×10−8 W/m²K45.6705 × 10^{−8}  W/m² K^{4}
Particle masses
kg u MeV/c²
Electron  9.1093897×10−319.1093897 × 10^{−31}  5.485798×10−45.485798 × 10^{−4} 0.5109991
Proton  1.6726231×10−271.6726231 × 10^{−27} 1.00727647 938.2723
Neutron  1.674955×10−271.674955 × 10^{−27} 1.008664924 939.5656
Deuteron  3.343586×10−273.343586 × 10^{−27} 2.013553 1875.6134
Conversion factors
1 eV  1.6021773×10−191.6021773 × 10^{−19}   J
1 u 931.4943  MeV/c²
 1.6605402×10−271.6605402 × 10^{−27}   kg
1 atomic unit 27.2114  eV
Δ x=0.96×2.998×108 m/s×3.57×10−10 s=0.1027 m.\Delta  x = 0.96 × 2.998 × 10^{8} \ m/s × 3.57 × 10^{−10} \ s = 0.1027 \ m.

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  0.1027×10−60.1027× 10^{−6} m  and the length of the track of a particle decaying by the strong interaction would be  0.1027×10−140.1027× 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.

14.3

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