Fermi Motion of Nucleons due to a Beam of Neutrinos

[vintagelover007]

Homework Statement

This is a homework assignment for my Modern Physics Class, I cannot find where to start in my textbook or class notes.

Relevant Equations

R = r0A^1/3
r_0=1.25*10^-15m

Stable nuclei have radii that are approximately given by the formula:
R = r0_A^1/3 Where r0 = 1.25 × 10−15m and A is the atomic mass number.

In many experiments of interest to modern particle physics, beams of neutrinos scatter from nucleons within the nucleus. Even though the nucleus is at rest, the nucleons inside it cannot be, because of the Heisenberg uncertainty principle.

The “Fermi momentum” they must carry has to be accounted for when modeling the scattering interaction. Approximating that the wave function of a given nucleon inside the nucleus has a 3D Gaussian wave function with σ = R:

a) Show that the mean momentum is: |pˆ|| = 4√ 2π σx/h bar integral dpp^3 exp [-2 p^2 R^2/h bar^2]

b) Evaluate this integral and calculate the mean momentum of the nucleons in a nucleus of 12^C.

 

[member 553083]

a) Using the definition of a Gaussian wave function, we can write the integral as: ∫dp^3exp[−2p^2R^2/hbar^2] = (2π hbar^2)^(3/2) ∫dp^3exp[−p^2/2]The integral is then equal to 1 since it is a normalized integral. Therefore, the mean momentum is:|pˆ|| = 4√2πσx/hbar (2π hbar^2)^(3/2) ∫dp^3exp[−p^2/2]|pˆ|| = 4√2πσx/hbarb) For 12^C, the atomic mass number A = 12 and the radius is given by R = r0_A^1/3 = 1.25 × 10−15m×12^1/3 = 2.42 × 10−14m. The mean momentum is then given by:|pˆ|| = 4√2πσx/hbar = 4√2π(2.42×10−14m)/(6.62×10−34m2s−1) = 0.072kgms−1