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# Nuclear Stability and Magic Numbers

Nuclear Stability is a concept that helps to identify the stability of an isotope. To identify the stability of an isotope or also known as the nuclei, you need to find the ratio of neutrons to protons.

### Introduction

A isotope is an element that has same atomic number but different atomic mass compared to the periodic table. Every element has a proton, neutron, and electron. The number of protons is equal to the atomic number, and the number of electrons is equal the protons, unless it is an ion.  To determine the number of neutrons in an element you subtract the atomic number from the atomic mass of the element. Atomic mass is represented as ($$A$$) and atomic number is represented as ($$Z$$) and neutrons are represented as ($$N$$).

$A-Z=N$

(atomic mass )-(atomic number)= number of neutrons

To determine the stability of an isotope you can use the ratio neutron/proton ($$N/Z$$). Also to help understand this concept there is a graph called the Belt of Nuclear stability. This graph is a plot of all the stable nuclei. The graph consists of a y-axis labeled neutrons, an x-axis labeled protons, and a nuclei. On the graph it is easier to determine where the alpha decay, beta decay, and positron emission or electron capture occurs.

### Determine N/Z ratio

Elements that have an atomic number (Z) lower than 20 are lighter and these elements' nuclei and have a ratio of 1:1. These elements prefer to have the same amount of protons and neutrons.

Example

Consider carbon-12 and oxygen-16

$\ce{^12_6C}$

and

$\ce{^12_8O}$

• Carbon: A-Z=N; 12-6= 6 ; 1:1 ratio (because there are 6 protons and 6 neutrons).
• Oxygen: A-Z= N; 16-8=8 1:1 ratio (because there are 8 protons and 8 neutrons).

Elements that have atomic numbers from 20 to 83 are heavy elements, therefore the ratio is different. The ratio is 1.5:1, the reason for this difference is because of the repulsive force between protons: the stronger the repulsion force, the more neutrons are needed to stabilize the nuclei.

### Belt of Stability

The graph of stable elements is commonly referred to as the Band (or Belt) of Stability. At the higher end of the band of stability lies alpha decay, below is positron emission or electron capture, above is beta emissions and elements beyond the atomic number of 83 are unstable radioactive elements.

Figure: Graph of isotope stability. Graph of isotopes by type of nuclear decay. Orange and blue nuclides are unstable, with the black squares between these regions representing stable nuclides. The unbroken line passing below many of the nuclides represents the theoretical position on the graph of nuclides for which proton number is the same as neutron number. The graph shows that elements with more than 20 protons must have more neutrons than protons, in order to be stable. Figure used with permission from Wikipedia

Alpha decay is located at the top of the plotted line, because the alpha decay decreases the mass number of the element in order to keep the isotope stable. This is done by using the element helium ($$He$$). An unstable isotope's protons are decreased by 2 and its neutrons are decreased by 4, and because the isotope was originally unstable before it went through alpha decay, the elements are still considered unstable.

Beta decay accepts protons so it changes the amount of protons and neutrons. the number of protons increase while neutrons decrease. To make things easier to understand think of the ratio of the isotope: there are too many neutrons compared to the number of protons therefore it is above the band of stability.

Positron emission and electron capture is when the isotope gains more neutrons. Positron emission and electron capture are below the band of stability because the ratio of the isotope has more protons than neutrons, think of it as there are too few protons for the amount of neutrons and that is why it is below the band of stability.

### Magic Numbers

Magic numbers are natural occurrences in isotopes and are stable. Below is a list of numbers of protons and neutrons; isotopes that have these numbers occurring in either the proton or neutron are stable. In some cases there the isotopes can consist of magic numbers for both protons and neutrons; these would be called double magic numbers. The double numbers only occur for isotopes that are heavier, because the repulsion of the forces between the protons.

The magic numbers:

• proton: 2, 8, 20, 28, 50, 82, 114
• neutron: 2, 8, 20, 28, 50, 82, 126, 184

Also, there is the concept that isotopes consisting a combination of even-even, even-odd, odd-even, and odd-odd are all stable. There are more nuclides that have a combination of even-even than odd-odd.

 Proton number (Z) Neutron Number # of stable Isotopes Even Even 163 Even Odd 53 Odd Even 50 Odd Odd 4

The four stable odd-odd nuclides:

• $$\ce{^2_1H}$$
• $$\ce{^{6}_3Li}$$
• $$\ce{^{10}_5B}$$
• $$\ce{^{14}_7N}$$

### Stable or Unstable?

Here is a simple chart that can help you decide if an element is stable:

Figure (left)-The contents of this figure has been taken from a lecture by Jim Hollister, LSC

### References

1. Olmsted III, John and Gregory M William. Chemistry Fourth Edition. John Wiley and Sons Inc:NJ, 2006.
2. Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, Jeffry D Madura. General Chemistry. Pearson Education Inc: NJ, 2007.

### Problems

1. Using the above chart state if this isotope is alpha-emitter, stable, or unstable:  a) 4020Ca   b) 5425Mn  c) 21084 Po
2. If the isotope is located above the band of stability what type of radioactivity is it? what if it was below?
3. Between elements bromide and carbon which is more stable when using magic numbers?
4. Name one of the isotopes that consist of odd-odd combination in the nuclei?

### Solutions

1)  a) Stable, because this Ca isotope has 20 neutrons, which is on of the magic numbers

b) Unstable, because there is an odd number (25 and 29) of protons and neutrons

c) Alpha-emitter, because Z=84, which follows rule/step one on the chart

2) Beta decay, positron emission, or electron capture

3) Carbon is stable

4)  Hydrogen-2, Lithium-6,  Boron-10, nitrogen-14

08:48, 9 Dec 2014

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