Understanding Cross Sections and Lifetimes: Questions 3 and 4

[MathematicalPhysicist]

Homework Statement

In the attached file, I need some help with questions 3 and 4.

Homework Equations

The Attempt at a Solution

For question 4, I get the next answers:
a. The flux is given by:
$$f=\frac{v_{\nu}\rho}{m}$$ where m is the mass of the iron nucleus, rho is the iron density and v_{\nu} is the nuetrino speed.

b. If I get this right then $$\frac{N_{events}}{N_{particles}}=\frac{\sigma_{\nu}f dx}{v_{\nu}}$$, and the number of scattered nuetrinos is N(x) times this fraction, i.e:
$$dN= \frac{\sigma_{\nu}f dx}{v_{\nu}}N$$.

c. easy as well, just identifying the above constant minus a sign with lambda.

d. Here is where I am findining it hard, I need to multiply sigma_{\nu} with some constant in order to eliminate the dimensions of energy in this cross section, I don't know how to do it, any hints as to how reconcile this?

The other questions are plug-and-calculate which aren't hard.

For question 3, I am not sure how to answer this question, any hints?

Thanks in advance.

 

[MathematicalPhysicist]

The attachment.