Everything about Liquid-drop Model totally explained
In
nuclear physics, the
semi-empirical mass formula (
SEMF), sometimes also called
Weizsäcker's formula, is a formula used to approximate the
mass and various other properties of an
atomic nucleus. As the name suggests, it's partially based on
theory and partly on
empirical measurements; the theory is based on the
liquid drop model, and can account for most of the terms in the formula, and gives a rough estimates for the values of the coefficients. It was first formulated in 1935 by
German physicist Carl von Weizsäcker, and although refinements have been made to the coefficients over the years, the form of the formula remains the same today.
This formula shouldn't be confused with the
mass formula of Weizsäcker's student
Burkhard Heim.
The formula gives a good approximation for atomic masses and several other effects, but doesn't explain the appearance of
magic numbers.
The liquid drop model and its analysis
The liquid drop model is a model in
nuclear physics which treats the
nucleus as a drop of incompressible nuclear fluid, first proposed by
George Gamow. The fluid is made of
nucleons (
protons and
neutrons), which are held together by the
strong nuclear force.
This is a crude model that doesn't explain all the properties of nuclei, but does explain the spherical shape of most nuclei.
It also helps to predict the binding energy of the nucleus.
Mathematical analysis of the theory delivers an equation which attempts to predict the binding energy of a nucleus in terms of the numbers of
protons and
neutrons it contains. This equation has five terms on its right hand side. These correspond to the cohesive binding of all the
nucleons by the
strong nuclear force, the
electrostatic mutual repulsion of the protons, a
surface energy term, an asymmetry term (derivable from the protons and neutrons occupying independent
quantum momentum states) and a pairing term (partly derivable from the protons and neutrons occupying independent
quantum spin states).
If we consider the sum of the following five types of energies, then the picture of a nucleus as a drop of incompressible liquid roughly accounts for the observed variation of binding energy of the nucleus:
Volume energy. When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. So, this nuclear energy is proportional to the volume.
Surface energy. A nucleon at the surface of a nucleus interacts with fewer other nucleons that one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.
Coulomb Energy. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy.
Asymmetry energy (also called
Pauli Energy). An energy associated with the
Pauli exclusion principle. If it wasn't for the Coulomb energy, the most stable form of nuclear matter would have
N=
Z, since unequal values of
N and
Z imply filling higher energy levels for one type of particle, while leaving lower energy levels vacant for the other type.
Pairing energy. An energy which is a correction term that arises from the tendency of proton pairs and neutron pairs to occur. An even number of particles is more stable than an odd number.
The formula
In the following formulae, let
A be the total number of
nucleons,
Z the number of
protons, and
N the number of
neutrons.
The mass of an atomic nucleus is given by
»
This is roughly
A/2 for light nuclei, but for heavy nuclei there's an even better agreement with
nature.
By substituting the above value of
Z back into
B one obtains the binding energy as a function of the atomic weight,
B(
A).
Maximizing
B(
A)/
A with respect to
A gives the nucleus which is most strongly bound, for example most stable. The value we get is
A=63 (
copper), close to the
measured values of
A=62 (
nickel) and
A=58 (
iron).
Further Information
Get more info on 'Liquid-drop Model'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://semi-empirical_mass_formula.totallyexplained.com">Semi-empirical mass formula Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
