Semi-empirical mass formula

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In nuclear physics, the semi-empirical mass formula (SEMF) is a formula used to approximate the mass and various other properties of an atomic nucleus. As the name suggests, it is partially based on theory and partly on empirical measurements; the theory suggests a form that the mass should take, and experiment provides the coefficients. It was first formulated in 1935 by German physicist Carl von Weizsäcker (note: this formula should not be confused with the mass formula of his student Burkhard Heim), and although refinements have been made to the coefficients over the years, the form of the formula remains the same today.

Contents

[edit] 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

m = Z m_{p} + N m_{n} - \frac{E_{B}}{c^{2}}

where mp and mn are the mass of a proton and a neutron, respectively, and EB is the binding energy of the nucleus. The semi-empirical mass formula states that the binding energy will take the following form:

E_{B} = a_{V} A - a_{S} A^{2/3} - a_{C} \frac{Z(Z-1)}{A^{1/3}} - a_{A} \frac{(A - 2Z)^{2}}{A} + \delta(A,Z)

Each of the terms in this formula has a theoretical basis, as will be explained below.

[edit] Volume term

The term aVA is known as the volume term. The volume of the nucleus is proportional to A, so this term is proportional to the volume, hence the name.

The basis for this term is the strong nuclear force. The strong force affects both protons and neutrons, and as expected, this term is independent of Z. Because the number of pairs that can be taken from A particles is \frac{A(A - 1)}{2}, one might expect a term proportional to A2. However, the strong force has a very limited range, and a given nucleon may only interact strongly with its nearest neighbors and next nearest neighbors. Therefore, the number of pairs of particles that actually interact is roughly proportional to A, giving the volume term its form.

[edit] Surface term

The term aSA2 / 3 is known as the surface term. This term, also based on the strong force, is a correction to the volume term.

The volume term suggests that each nucleon interacts with a constant number of nucleons, independent of A. While this is very nearly true for nucleons deep within the nucleus, those nucleons on the surface of the nucleus have fewer nearest neighbors, justifying this correction.

If the volume of the nucleus is proportional to A, then the radius should be proportional to A1 / 3 and the surface area to A2 / 3. This explains why the surface term is proportional to A2 / 3.

[edit] Coulomb term

The term a_{C} \frac{Z(Z-1)}{A^{1/3}} is known as the Coulomb or electrostatic term.

The basis for this term is the electrostatic repulsion between protons. To a very rough approximation, the nucleus can be considered a sphere of uniform charge density. The potential energy of such a charge distribution can be shown to be

\frac{3}{5} \frac{1}{4 \pi \epsilon_{0}} \frac{Q^{2}}{R}

where Q is the total charge and R is the radius of the sphere. Identifying Q with Ze, and noting as above that the radius is proportional to A1 / 3, we get close to the form of the Coulomb term. However, because electrostatic repulsion will only exist for more than one proton, Z2 becomes Z(Z − 1). Because the nucleus is not really a sphere of uniform charge density, we cannot to any degree of accuracy calculate aC directly from the equation above; it must be measured experimentally as with the other coefficients.

[edit] Asymmetry term

Illustration of basis for asymmetric term

The term a_{A} \frac{(A - 2Z)^{2}}{A} is known as the asymmetry term. The theoretical justification for this term is more complex, but it is based on the asymmetry between the number of protons and neutrons. Note that as A = N + Z, the parenthesized expression can be rewritten as (N − Z). The form (A − 2Z) is used to keep the dependence on A explicit, as will be important for a number of uses of the formula.

The Pauli exclusion principle states that no two fermions can occupy exactly the same quantum state. At a given energy level, there are only finitely many quantum states available for particles. What this means in the nucleus is that as more particles are "added", these particles must occupy higher energy levels, increasing the total energy of the nucleus (and decreasing the binding energy). Note that this effect is not based on any of the fundamental forces (gravitational, electromagnetic, etc.), only the Pauli exclusion principle.

Protons and neutrons, being distinct types of particles, occupy different quantum states. One can think of two different "pools" of states, one for protons and one for neutrons. Now, for example, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy than the available states in the proton pool. If we could move some particles from the neutron pool to the proton pool, in other words change some neutrons into protons, we would significantly decrease the energy. The imbalance between the number of protons and neutrons causes the energy to be higher than it needs to be, for a given number of nucleons. This is the basis for the asymmetry term.

The actual form of the asymmetry term is to some extent a guess, but it is also one that provides a good fit for the experimental data. The term should be dependent on the absolute difference | N − Z | , and the form (A − 2Z)2 is simple and differentiable, which is important for certain applications of the formula. In addition, small differences between Z and N are not "punished" too much. The A in the denominator can be intuitively thought of as a normalizing factor, which reflects the fact that a given difference | N − Z | is less significant for larger values of A.

[edit] Pairing term

The term δ(A,Z) is known as the pairing term (possibly also known as the pairwise interaction). This term captures the effect of spin-coupling. It is given by:

\delta(A,Z) = \begin{cases} +\delta_{0} & Z,N \mbox{ even } (A \mbox{ even}) \\ 0 & A \mbox{ odd} \\ -\delta_{0} & Z,N \mbox{ odd } (A \mbox{ even})\end{cases}

where

\delta_{0} = \frac{a_{P}}{A^{1/2}}

It is known that a pair of particles having opposite spin has slightly lower energy than the sum of the particles considered separately. If both Z and N are even, all nucleons will exhibit spin-coupling. If both are odd, there will be two particles that do not exhibit spin-coupling. The case of one of N, Z even, the other odd, is an intermediate case.

The factor A1 / 2 is not easily explained, and like the form of the asymmetry term, is based more on empirical evidence.

[edit] Calculating the coefficients

The coefficients are calculated by fitting to experimentally measured masses of nuclei. Their values can vary depending on how they are fitted to the data. Several examples are as shown below, with units of mega-electronvolts (MeV):

Least-squares fit[citation needed] Wapstra Rohlf
aV 15.8 14.1 15.75
aS 18.3 13 17.8
aC 0.714 0.595 0.711
aA 23.2 19 23.7
aP 12 n/a n/a
δ (even-even) n/a -33.5 +11.18
δ (odd-odd) n/a +33.5 -11.18
δ (even-odd) n/a 0 0
  • Wapstra: Atomic Masses of Nuclides, A. H. Wapstra, Springer, 1958
  • Rohlf: Modern Physics from a to Z0, James William Rohlf, Wiley, 1994

The semi-empirical mass formula provides a good fit to heavier nuclei, and a poor fit to very light nuclei, especially 4He. This is because the formula does not consider the internal shell structure of the nucleus. For light nuclei, it is usually better to use a model that takes this structure into account.

[edit] References

  1. R.Freedman, H.Young (2004), University Physics with Modern Physics, 11th international edition, Sears and Zemansky, 1633-4. ISBN 0-8053-8768-4.

[edit] External links

[edit] See also

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