10 Nuclear Models-2

Sanjay Kumar Chamoli

1. Applications of Liquid Drop Model

To go further, we are interested in the deviations between the experimental measured values and the values calculated from liquid drop model for the binding energies, and in understanding these deviations in terms of microscopic models. The deviation between the measured values and the Bethe-Weizsacker values for binding energy per nucleon is given in fig. 1. While the fit is generally good, periodic strong deviations are observed at certain proton or neutron numbers. These deviations indicate additional stability and provide evidence for shell closures at nucleon numbers 2, 8, 20, 28, 50, 82 and 126 (neutrons).

Fig. 1: Difference between measured masses values and those predicated by the Bethe-Weizsacker formula for Binding energy per nucleon (B.E./A

1.1 Interpretation of Liquid Drop Mass Formula

The liquid drop mass equation leads to the following general conclusions

All nuclei except three to four exceptions are most stable in nature up to lead. Nuclei near ⁵⁶Fe are the most stable in Nature, which can be proved by differentiating semi-empirical mass equation with respect to Z and A and setting the result equal to zero to obtain the minimum. This is the point at which the competition between surface energy losses and Coulomb energy losses is balanced, corresponding to the peak in the average binding energy curve. Therefore, all nuclei are thermodynamically driven to ⁵⁶Fe.
Nuclei with low A lose binding energy primarily due to the surface energy term; i.e. as nuclei become smaller, a larger fraction of the nucleons are on the surface. By adding two light nuclei together in a nuclear fusion reaction, a more stable system is formed and energy is released. This is the principle upon which stars generate energy and is the basis for the nuclear fusion reactor.
Nuclei with high A lose binding due to their large Coulomb energy. In heavier nuclei due to the presence of large number of protons large coulomb repulsion exist between them and coulomb energy dominates over surface energy and splitting take place by releasing excess energy. Thus, splitting a heavy nucleus in a nuclear fission reaction leads to more stable products and energy is released.
The symmetry term favors N/Z = 1. So all the stable nuclei lie on or near a line called stability line in Z, N plane. For light nuclei this is observed up to Ca40. However, for heavier nuclei neutron and proton ratio is found to be greater than unity i.e. N/Z > 1. This trend is observed due to additional number of neutrons to balance with the Coulomb energy; i.e. larger A for a given Z lowers the Coulomb energy loss.
Except few exceptions, all stable nuclei have N = Z, Z+1, or Z+2 upto mass A = 35 & thereafter increases faster than Z until in lead N ≈ 1.5 Z.
Nuclei with even numbers of neutrons and protons have higher binding energy relative to those with an odd nucleon.

2. Stability of Nucleus

In the plot of N versus Z graph for the most stable nuclei shows the divergence from N = Z, as preferred by the symmetry term in the liquid drop (LD) mass equation. To trace the evolution of this plot, it is useful to examine the prediction of the mass formula for individual isobars A, which are connected via beta decay. In order to find out the stability of the nucleus, we can take the first derivative with respect to Z to calculate the optimal Z such that mass is minimum.

From the Bethe-Weizsacker Mass formula

This is the equation of parabola in Z for given A, and the minimum of the parabola corresponds to the most stable nucleus for a given isobar Z, which can be obtained by differentiating above and setting the result equal to zero. These minima define the ‘valley of beta stability’. Where

2.1 Implications

From the above equation we can conclude that the most stable nuclei of various mass numbers A is determined by the value of Z₀. The deviation of stability line from the condition N=Z or Z= A/2 is due to competition between the Coulomb energy, which favors Z₀ < A/2, & the Asymmetry energy which favors Z₀ = A/2, and is represented in fig. 2.

Fig. 2: Deviation of Stability line with increasing number of protons

Because of the pairing energy term δ in the mass equation, the solution fall into two categories according to whether even-A or odd-A nuclei is taken into consideration. For odd-A nuclei the paring term (δ) is zero, leading to a single parabola for both e-o and o-e nuclei shown in fig. 3(a). The important result for this case is that there is only one beta-stable isotope for odd-A nuclei.

For even-A nuclei, the pairing term can be either +1 for e-e nuclei or -1 for o-o nuclei. This yields two parabolas for the same mass number shown in fig. 3 (b), one for even Z (e-e) and the other for odd Z (o-o). Since the beta decay of an e-e nucleus produces an o-o nucleus, the decay chain alternates between the two parabolas. For even isobars, since the values of can be positive or -negative, so the mass equation gives two parabolas, differing in mass by 2δ. The important result for this case is that even-A nucleus can have up to three beta-stable isobars, depending on the relative positions of the two parabola.

Fig. 3: Mass parabolas for Even and Odd A. Z0 is showing position of minima, corresponding to most stable isobar.

In the above fig. 3 (a) is shown the plot of M (A, Z) against Z for odd A isobars with A= 135. This is a parabola for which the lowest point is at Z0= 55.7. The stable isobar that is actually observed at this mass number is Ba¹³⁵ for which Z= 56. The nuclides falling on either side of the stable isobar are all unstable. Those on the lower Z side (Z <56) are β^– active while those on the higher side (Z > 56) are β⁺ active or electron capturing. Each of these nuclei undergoes β^– transformation into the product nucleus with Z one unit higher or lower respectively as shown in the figure. Such transformation goes on step by step till the stable end product is reached.

Fig. 3 (b) shows the two mass parabolas for the even A isobars with A = 102. The upper one is for odd Z, odd N isobars, while the lower one is for the even Z, even N isobars. The most stable isobar in this case falls on the lower parabola. On using the equation for Z₀ to calculate the minimum value of Z₀ protons for stable isobar, we obtained Z₀ = 44.7. Actually a stable nuclide 102Ru at Z = 44 is observed at this mass number. In addition to this, another stable e-e nuclide ¹⁰²Pd (Z =46) also exist at this mass number A. The two stable isobars differ in Z by two units. The o-o isobar ¹⁰²Rh with Z = 45 between these two falls on the upper parabola and has an atomic mass greater than those of either of the above two. Hence ¹⁰²Rh is not stable. It shows both β⁺ and β^– activities. β⁺ emission transforms it to ¹⁰²Ru while by – emission it transforms to ¹⁰²Pd. Fig. 3(b) also show two other o-o isobars ¹⁰²Tc with Z= 43 and ¹⁰²Ag with Z = 47 on the upper parabola. Their atomic masses are higher than those of the neighboring e-e isobars ¹⁰²Ru and ¹⁰²Pd respectively. Hence none of those latter two isobars can be β⁻ active. β- transformation changes Z by one unit.

Depending on the curvature of the parabolas & separation 2δ, there can be several stable even-even isobars for which Z differs by two units. There exist a possibility that for certain odd-odd nuclei both conditions are fulfilled so both β⁻ & β⁺decay from the identical nuclide are possible and do indeed occur as shown in fig. 4.

Fig. 4: Figure indicating stable isobar of odd-A nuclei

2.2 Rules of decay for even- A nuclei

There are certain rules by which an even-A nuclei can decay

Even-A & Even-Z nuclei have a high B.E. due to pairing term, whereas odd-Z nuclei have a lower B.E. due to opposite contribution from this term. Thus there are two curves of isobar mass against Z & alternate Z lie on different curves. The odd nuclei with Z = 43 can decay by Electron capture (EC) to Z = 42 or by negative beta β^– decay to Z = 44. The important conclusion of this results is that there are two stable isobars for A = 100, namely Z = 42 & 44, which is true as shown in fig. 5.

Figure 5: Figure showing two stable isobar corresponding to even-A nuclei

So we conclude from the above discussion that there can be two even-even stable isobars (A even) for which Z differs by two units.

Sometimes there is a possibility that even-A nucleus leaves an odd-odd isobar in an excited state, so there a probability that it energetically be able to decay by all β^– and β⁺ (or E.C.) modes. (⁴⁰K₁₉) as shown in fig. 6.

Fig. 6: Decay probability of even-A nucleus ⁴⁰K₁₉. It shows that ⁴⁰K₁₉ can decay by all three modes

Summary

The Bethe–Weizsacker mass formula is a phenomenological understanding of nuclear binding energies as function of A, Z and N. and explains the experimental B.E./A quite well. It explains the valley of stability; and explains the energetics of radioactive decays, fission and fusion. The Bethe–Weizsacker mass formula also define the limit of stability against alpha-decay and spontaneous fission. The Coulomb term and the Asymmetry energy term play an important role in deciding the stability of nuclei. The odd-A nuclei can have only one mass parabola due to paring term is zero, whereas even-A nuclei have two mass parabolas as pairing energy tern can take two values either –δ or +δ.

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References:

Introduction to Nuclear Physics – by Keneth S Krane.
Introductory Nuclear Physics – by Samuel S M Wong.
Nuclear Physics – by R R Roy & B P Nigam.
Elementary Nuclear Theory by Hans A. Bethe and Phillip Morrison.
Carl Friedrich von Weizsäcker: Pioneer of Physics, Philosophy, Religion … edited by Ulrich Bartosch
Hans Bethe and His Physics By Gerald Edward Brown, Chang-Hwan Lee
Introduction to Nuclear Physics, 2^nd Edition, W.N.Cottingham & D.A. Greenwood.
Concept of Nuclear Physics by B L Cohen, McGraw Hill.
Nuclear Physics ; an Introduction by S.B. Patel.
Exotic Nuclear Excitation by S.C. Pancholi
Nuclear spectroscopy Part B, by Fay Ajzenberg- Selove
Basic Ideas & Concepts in Nuclear Physics – by K Heyde
5. Nuclear Physics by Irving Kaplan, Narosa Publishing House.

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Did you know ?

The Liquid Drop Model examines the global properties of nuclei, such as energetics, binding energies, sizes, shapes and nucleon distributions.
This model assumes that all nucleons are alike (other than charge).
The basic assumption of the Liquid Drop Model is that the nucleus is a charged, nonpolar liquid drop held together by the nuclear force.
In the simplest case, the chemical analogy would be a droplet of composed of nonpolar molecules such as CCl4 or iso-pentane held together by Vander Waal’s attraction.\
There are significant differences between a classical liquid drop and a nucleus which must be accounted for in the model.
The nucleus has a limited number of par particles ( ~ <270) compared to chemical systems (~10 23 ). The net result is that there is a much larger fraction of nucleons on the surface relative to those in the bulk for nuclei compared to chemical systems.
The nucleus is a two-component system composed of neutrons and protons. Protons in the nucleus carry positive electric charge. Therefore, they lose energy due to mutual Coulomb repulsion. An analogous chemical case would be an hypothetical cluster of Xe and Xe ions held together by much stronger than normal Van de Waal’s forces. Microscopic properties such as shell structure are not included in the model
Due to the effect of the liquid drop model the nuclei near 56Fe are the most stable in Nature. This happens due to the competition between surface energy losses and Coulomb energy losses is balanced, Therefore, all nuclei are thermodynamically driven to ⁵⁶Fe
Nuclei with low A lose binding energy primarily due to the surface energy term; i.e. as nuclei become smaller, a larger fraction of the nucleons are on the surface. By adding two light nuclei together in a nuclear fusion reaction, a more stable system is formed and energy is released. This is the principle upon which stars generate energy and is the basis for the nuclear fusion reactor, which someday may become a major source of commercial electricity.
The liquid drop model successfully explained the fission process in nuclei which is the energy source for present-day nuclear reactors. Nuclei with high A lose binding due to their large Coulomb energy. Thus, splitting a heavy nucleus in a nuclear fission reaction leads to more stable products and energy is released.
The symmetry term favors N/Z = 1. For light nuclei this is observed up to ⁴⁰Ca. However, for heavier nuclei N/Z > 1. This trend is due to a balance with the Coulomb energy; i.e. larger A for a given Z lowers the Coulomb energy loss. Nuclei with even numbers of neutrons and protons gain additional binding relative to those with an odd nucleon.
Most naturally occurring nuclides are stable while the others are known as radioactive with sufficiently long half-lives to occur primordially.
If the half-life of a nuclide is comparable to, or greater than, the Earth’s age (4.5 billion years), a significant amount will have survived since the formation of the solar system and then it is called primordial. It will then contribute in that way to the natural isotopic composition of a chemical element.
Primordially present radioisotopes are easily detected with half-lives as short as 700 million years (e.g., ²³⁵U). This is the present limit of detection, as shorter-lived nuclides have not yet been detected undisputedly in nature.
Many naturally occurring radioisotopes exhibit still shorter half-lives than 68 million years, but they are made freshly, as daughter products of decay processes of primordial nuclides (for example, radium from uranium).
Some isotopes that are classed as stable (i.e. no radioactivity has been observed for them) are predicted to have extremely long half-lives (sometimes as high as 1018 years or more). If the predicted half-life falls into an experimentally accessible range, such isotopes have a chance to move from the list of stable nuclides to the radioactive category, once their activity is observed. E.g. 209Bi and 180W were formerly classed as stable, but have been recently (2003) found to be alpha-active. However, such nuclides do not change their status as primordial when they are found to be radioactive.
Most stable isotopes in the earth are believed to have been formed in processes of nucleosynthesis either in the Big Bang, or in generations of stars that preceded the formation of the solar system.
Some stable isotopes also show abundance variations in the earth as a result of decay from long-lived radioactive nuclides. These decay-products are termed as radiogenic, in order to distinguish them from the much larger group of ‘non-radiogenic’ isotopes.
The island of stability reveal a number of long-lived or even stable atoms that are heavier (and with more protons) than lead.

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