Nuclear Fission: The Basic Process — Question 8

Question 8

A neutron multiplicity distribution has both a mean and a spread

For an illustrative ensemble of independent fissions, the emitted prompt-neutron number n is 2 with probability .30, 3 with probability .50, and 4 with probability .20. No subsequent neutron reactions are included.

1. Calculate the mean and variance of n.

2. For 100 independent events, find the mean total neutron count and its standard deviation. Explain why the mean need not be an integer.

Question 8 - Solution

A neutron multiplicity distribution has both a mean and a spread

(a) Method and calculation

$$\boxed{\langle n\rangle=2(.30)+3(.50)+4(.20)=2.90}.$$$$\boxed{\operatorname{Var}(n)=4(.30)+9(.50)+16(.20)-(2.90)^2=.490}.$$

(b) Interpretation and further calculation

$$\boxed{\langle N\rangle=100(2.90)=290,\quad\sigma_N=\sqrt{100(.490)}=7.00}.$$$$\overline n=\frac{\sum_i n_i}{M}\to2.90\quad(M\to\infty),\qquad\boxed{n_i\in\{2,3,4\}}.$$

Physical check

The standard deviation describes ensemble fluctuations, not an uncertainty interval that constrains every individual realization.