Quantum Kinetic Energy of Neutrons, Protons and Electrons

[gennarakis]

Hi there,

I have a problem to solve in Cosmology which says:

"Write the formulas for the quantum kinetic energy of neutrons, protons and electrons as well as the formula for the gravitational energy for a neutron star that is comprised of free neutrons, protons and electrons in a ratio of N_n : N_p : N_e = 93,4 : 6,6 : 6,6 (N_ν=N_n+N_n=1,8x10⁵⁷)

The numerical coefficients should be these for homogeneous density."


Attempt for solution

I find in the book "From Quarks to Quasars" by E.N.Economou that the total quantum kinetic energy of neutrons is 92,14% of the total, U_k,tot=U_k,n/0,9214=(α_κ/0,9214)ħ²N^5/3/(m_nR²). The coefficients are a_κ=1,1.

This equation is derived from the total kinetic energy of N fermionic particles (s=1/2) that are similar and non relativistic.

And my question is:

How can this equation be used for neutrons (and protons) which are bosons and don't follow Fermi's exclusion principle (where this equation is derived from)?

 

[Garlic]

How can this equation be used for neutrons (and protons) which are bosons and don't follow Fermi's exclusion principle (where this equation is derived from)?

Protons and neutrons are fermions with spin 1/2 h/2π, and they are subjects to the Pauli exclusion principle.
All baryons are fermions, without exception.

 

[gennarakis]

Protons and neutrons are fermions with spin 1/2 h/2π, and they are subjects to the Pauli exclusion principle.
All baryons are fermions, without exception.

I was confused for a second though I see my question is probably of lack of knowledge..but all bosons must have an integer spin, right? so, how baryons which are comprised of three quarks and hence have an integer spin are fermions and not bosons?

 

[Garlic]

I was confused for a second though I see my question is probably of lack of knowledge..but all bosons must have an integer spin, right? so, how baryons which are comprised of three quarks and hence have an integer spin are fermions and not bosons?

Neutrons and protons have half-integer spin (1/2 h/2π).
This can be explained with the fact that individual quarks have spin 1/2 h/2π, and therefore baryons have ±1/2 or ±3/2 spin (as quarks can have different positions).
Mesons (like pions) however, have always integer spins.

 

[gennarakis]

Neutrons and protons have half-integer spin (1/2 h/2π).
This can be explained with the fact that individual quarks have spin 1/2 h/2π, and therefore baryons have ±1/2 or ±3/2 spin (as quarks can have different positions).
Mesons (like pions) however, have always integer spins.

Thanks a lot! You are right...I confused the electric charge with the spin of the particle and I couldn't see how three quarks with 2/3 and -1/3 spins cannot be integers...

 

[Garlic]

Thanks a lot! You are right...I confused the electric charge with the spin of the particle and I couldn't see how three quarks with 2/3 and -1/3 spins cannot be integers...

I'm glad I could help.