# Neutron's crow flight distance & 2° moment of a distribution

#### dRic2

Gold Member
Hi, I'm looking for a simple explanation of the meaning of the crow flight distance and why it is defined as the second moment of a probability distribution:
$$\bar r^2 = \int r^2 p(r)dr$$
Where $p(r)$ is the probability that a neutron is absorbed in the interval $dr$ near $r$. And what exactly is the meaning of the second moment of a probability distribution ?

Thanks
Ric

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#### rpp

I don't think r^2 is the "crow flight distance".

Are you referring to the older Lamarsh book? (This is the only nuclear engineering book that I know of that has crow flight distance).
He talks about the crow flight distance, but he uses r^2 because it is a quantity that is part of the "fermi age" theory.

#### dRic2

Gold Member
Yes, I realized later that $\bar r$ should be the crow flight distance and not $\bar {r^2}$, but I still do not understand why he uses $\bar {r^2}$. In fact $\sqrt{ \bar {r^2}} \neq \bar r$

PS: yes, I'm reading Lamarsh book

#### Astronuc

Staff Emeritus
I believe Lamarsh was using the formula (Equation 6-105) for the second moment of the distribution function, p(r), to obtain an expression in terms of τ, so that one would see that τ is equal to $\frac{1}{6}\bar{r^2}$.

It's been about 40 years since I had to work those equations, and I think we had to show that $\bar{r}^2 \neq \bar{r^2}$, the former being the square of the first moment of the distribution function.

$$\bar{r^2} = \int r^2 p(r)dr$$ and $$\bar r^2 = {\Big(\int r p(r)dr\Big)}^2$$

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#### dRic2

Gold Member
I believe Lamarsh was using the formula (Equation 6-105) for the second moment of the distribution function, p(r), to obtain an expression in terms of τ, so that one would see that τ is equal to 16¯r216r2¯\frac{1}{6}\bar{r^2}.
Yes, but I thought that maybe it could have a deeper meaning...

"Neutron's crow flight distance & 2° moment of a distribution"

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