Semi Empirical Mass Formula and Neutron Stars

In summary, the Semi Empirical Mass Formula (SEMF) provides a theoretical framework for understanding the binding energy of atomic nuclei, which is crucial in nuclear physics. When applied to neutron stars, SEMF helps explain their structure and stability by accounting for the contributions of nuclear forces, surface energy, and symmetry energy. Neutron stars, composed predominantly of neutrons, exhibit unique properties due to extreme densities, and SEMF aids in modeling the behavior of matter under such conditions, offering insights into phenomena like gravitational collapse and neutron degeneracy pressure.
  • #1
city2113
2
1
Homework Statement
Using the liquid drop model but adding a term for gravity to apply it to a neutron star, with only a large number of neutrons, rewrite the binding energy equation for this case
Relevant Equations
Semi Empirical Mass Formula: B(A,Z) = avA - asA^(2/3)-aa(A-2Z)^2/A+ap(delta/A^(1/2))
Gravity Term to add: (3/5)((G/ro)A^2*mn^2)/A^(-1/3)
Sorry, the equations are messy. I already know the answer to the actual homework problem, but I don't really know why certain terms are neglected.
I know that the equation will only include the volume term, symmetry term and the extra added gravity term. I just want to understand why

I know that the coulomb term is ignored because if you have only neutrons, there is no charge to consider.
I also know that the surface term and the pairing term become negligible at very large N values, but I don't know why.

Could someone explain why the surface and pairing terms get neglected in this case?
 
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  • #2
Hi @city2113 and welcome to PF.

I don’t understand enough to know why the pairing term becomes negligible. But see if this helps with the surface term...

The volume term depends on the total number of nucleons and the number of (very short range strong nuclear force) interactions of each nucleon with its neighbours.

The surface term is simply a correction to the volume term to allow for the fact that surface nucleons have fewer neighbours than the internal nucleons.

Suppose there are ##A## nucleons and ##f## is the fraction of the nucleons which are at the surface.

What do you think happens to ##f## as ##A## gets big?
 
  • #3
Ohh ok, so as the number of nucleons (A) gets bigger, the fraction of nucleons on the surface would be smaller compared to the total number of nucleons. Once A becomes really bug the effect of the ones on the surface having fewer neighbors becomes so small that it's negligible because there are far more in the middle.

Thank you, that helps a lot!
 
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FAQ: Semi Empirical Mass Formula and Neutron Stars

What is the Semi Empirical Mass Formula (SEMF) and how is it used?

The Semi Empirical Mass Formula (SEMF), also known as the Weizsäcker formula, is a mathematical model used to approximate the binding energy of atomic nuclei. It accounts for various factors such as volume energy, surface energy, Coulomb energy, asymmetry energy, and pairing energy. This formula helps in understanding the stability of nuclei and predicting nuclear masses, which is crucial in nuclear physics and astrophysics.

How does the SEMF relate to neutron stars?

The SEMF provides insights into the properties of atomic nuclei, which are the building blocks of neutron stars. Neutron stars are incredibly dense objects primarily composed of neutrons, and understanding the nuclear binding energy helps in modeling the behavior of matter under extreme conditions. The SEMF aids in predicting the equation of state (EOS) of neutron star matter, which is essential for understanding their structure and evolution.

What role does the asymmetry energy term in the SEMF play in neutron stars?

The asymmetry energy term in the SEMF accounts for the energy difference between protons and neutrons in a nucleus. In neutron stars, where the neutron-to-proton ratio is extremely high, the asymmetry energy becomes significant. It influences the stability and properties of neutron-rich matter, affecting the overall equation of state and the neutron star's characteristics, such as its radius, mass, and cooling behavior.

How does the SEMF help in determining the maximum mass of neutron stars?

The SEMF contributes to the understanding of the nuclear forces at play within neutron star matter. By incorporating the SEMF into the equation of state for neutron star matter, scientists can predict the maximum mass that a neutron star can have before collapsing into a black hole. This limit, known as the Tolman–Oppenheimer–Volkoff (TOV) limit, is crucial for astrophysical observations and theories regarding the life cycle of stars.

Can the SEMF be applied directly to neutron star matter?

While the SEMF provides valuable insights into nuclear properties, it is not directly applicable to neutron star matter due to the extreme conditions present in neutron stars, such as high densities and pressures. However, it serves as a foundational tool for developing more sophisticated models and equations of state that can accurately describe the behavior of matter in neutron stars. These advanced models build upon the principles of the SEMF to better understand neutron star physics.

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