Question

The half-life of "free" neutrons (those not bound inside a nucleus) is 10 minutes. Suppose that 100,000 neutrons are created in a reactor each second; they have β-0.8 and are directed toward an experimental target some distance Ax away. Detectors monitoring the target reveal that 60,000 neutrons arrive there per second. What is ΔΧ7(5 points) Do you think this problem is realistic? (3)

Answer 1

Proper time elapsed

$T=\frac{1}{\gamma}\frac{\Delta x}{v}$

Decay constant

$\lambda=\frac{\ln2}{t_{1/2}}$

Thus, according to the problem statement

$\lambda T=\ln\left (\frac{N_0}{N} \right )$

(which is the radioactive decay law)

$\Delta x=\frac{v\gamma t_{1/2}}{\ln2}\ln\left (\frac{N_0}{N} \right )=\frac{c\beta}{\sqrt{1-\beta^2}} t_{1/2}\log_2\left (\frac{N_0}{N} \right )$

Put all the values to get

$\Delta x=\frac{3\times10^8\times0.8}{\sqrt{1-0.8^2}}\times 10\times60\log_2\left (\frac{100000}{60000} \right )=1.77\times10^{11}\textup{ m}$

which is obviously a nonsense result. It is more than the distance of sun from us!!