Contents

- 1 Atomic Mass Number
- 2 Atomic Number, Neutron Number and Nuclear Stability
- 3 Atomic Mass Number – Does it conserve in a nuclear reaction?
- 4 Atomic Mass Number and Neutron Slowing Down
- 5 Atomic Mass Number and Nuclear Radius
- 6 Atomic Mass Number and Nuclear Fission
- 7 Atomic Mass Number and Atomic Mass (Isotopic Mass)
- 8 See also:

## Atomic Mass Number

**The atom** consist of a small but massive **nucleus **surrounded by a cloud of rapidly moving **electrons**. The nucleus is composed of **protons and neutrons**. The total number of protons and neutrons in the nucleus of an atom is called the **atomic mass number** (or the mass number) of the atom and is given the symbol **A**.

Neutron number plus atomic number equals atomic mass number: **N+Z=A**. The difference between the neutron number and the atomic number is known as the **neutron excess**: D = N – Z = A – 2Z.

The **chemical properties** of the atom are determined by the **number of protons**, in fact, by number and arrangement of electrons. The configuration of these electrons follows from the principles of quantum mechanics. The number of electrons in each element’s electron shells, particularly the outermost valence shell, is the primary factor in determining its chemical bonding behavior. In the periodic table, the elements are listed in order of increasing atomic number Z.

The **nuclear properties** (atomic mass, nuclear cross-sections) of the element are determined by the **number of protons** (atomic number) and **number of neutrons **(neutron number). For example, actinides with odd neutron number are usually fissile (fissionable with slow neutrons) while actinides with even neutron number are usually not fissile (but are fissionable with fast neutrons). Heavy nuclei with an even number of protons and an even number of neutrons are (due to Pauli exclusion principle) very stable thanks to the occurrence of ‘paired spin’. On the other hand, nuclei with an odd number of protons and neutrons are mostly unstable.

See also: Weizsaecker Formula Asymmetry and Pairing term

**Atomic mass number** determines especially the atomic mass of atoms. The mass number is different for each different isotope of a chemical element.

The **mass number** is written either after the element name or as a superscript to the left of an element’s symbol. For example, the most common isotope of carbon is carbon-12, or ^{12}C.

## Atomic Number, Neutron Number and Nuclear Stability

**Nuclear Stability** is a concept that helps to identify the stability of an isotope. To identify the stability of an isotope it is needed to find the ratio of neutrons to protons. 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 chart of the nuclides, known as a Segre chart. This chart shows a plot of the known nuclides as a function of their atomic and neutron numbers. It can be observed from the chart that there are **more neutrons than protons** in nuclides with **Z greater** than about 20 (Calcium). These **extra neutrons** are necessary for stability of the heavier nuclei. The excess neutrons act somewhat like nuclear glue.

See also: Livechart – iaea.org

Atomic nuclei consist of protons and neutrons, which attract each other through **the nuclear force**, while protons repel each other via **the electric force** due to their positive charge. These two forces compete, leading to various stability of nuclei. There are only certain combinations of neutrons and protons, which forms **stable nuclei**.

**Neutrons stabilize the nucleus**, because they attract each other and protons , which helps offset the electrical repulsion between protons. As a result, as the number of protons increases, **an increasing ratio of neutrons to protons is needed** to form a stable nucleus. If there are too many or too few neutrons for a given number of protons, the resulting nucleus is not stable and it undergoes radioactive decay. **Unstable isotopes**decay through various radioactive decay pathways, most commonly alpha decay, beta decay, or electron capture. Many other rare types of decay, such as spontaneous fission or neutron emission are known. It should be noted that all of these decay pathways may be accompanied by **the subsequent emission of gamma radiation**. Pure alpha or beta decays are very rare.

## Atomic Mass Number – Does it conserve in a nuclear reaction?

In general, the **atomic mass number is not conserved** in nuclear reactions.

In analyzing nuclear reactions, we apply the **many conservation laws**. **Nuclear reactions** are subject to classical **conservation laws for charge, momentum, angular momentum, and energy **(including rest energies). Additional conservation laws, not anticipated by classical physics, are are **electric charge**, **lepton number and baryon number**. Certain of these laws are obeyed under all circumstances, others are not. We have accepted conservation of energy and momentum. In reactor physics (non-relativistic physics), we assume that the number of protons (**the atomic number**), the number of neutrons (**the neutron number**) and its sum (**the atomic mass number**) are usually separately conserved. We shall find circumstances and conditions in which this rule is not true. Where we are considering non-relativistic nuclear reactions, it is essentially true. However, where we are considering **relativistic nuclear energies** or those involving the **weak interactions **(e.g. in beta decay the atomic number is not conserved), we shall find that these principles must be extended.

Instead of mass number conservation, physicists define the **baryon number, **which **is a conserved quantum number** in all particle reactions.

**Baryon number** is a generalization of **nucleon number**, which is conserved in nonrelativistic nuclear reactions and decays. The **law of conservation of baryon number**states that:

*The sum of the baryon number of all incoming particles is the same as the sum of the baryon numbers of all particles resulting from the reaction.*

For example, the following reaction (proton-antiproton pair production) does conserve B and does occur if the incoming proton has sufficient energy (the threshold energy = 5.6 GeV):

As indicated, B = +2 on both sides of this equation.

From these and other reactions, the conservation of baryon number has been established as a basic principle of physics.

This principle provides basis for the **stability of the proton**. Since the proton is the lightest particle among all baryons, the hypothetical products of its decay would have to be non-baryons. Thus, the decay would violate the conservation of baryon number. It must be added some theories have suggested that protons are in fact unstable with very long half-life (~10^{30} years) and that they decay into leptons. There is currently no experimental evidence that proton decay occurs.

## Atomic Mass Number and Neutron Slowing Down

**The moderator**, which is of importance in thermal reactors, is used to moderate, that is, **to slow down**, neutrons from fission to thermal energies. The probability that fission will occur depends on incident neutron energy. Physicists calculate with fission cross-section, which determines this probability.

Nuclei with low mass numbers are most effective for this purpose, so the moderator is always a low-mass-number material. Commonly used moderators include regular (light) water (roughly 75% of the world’s reactors), solid graphite (20% of reactors) and heavy water (5% of reactors). Beryllium and beryllium oxide (BeO) have been used occasionally, but they are very costly. Low-mass number materials are effective due to high **logarithmic energy decrement per collision (ξ)** as a key material constant describing energy transfers during a neutron slowing down.

## Atomic Mass Number and Nuclear Radius

**Typical nuclear radii** are of the order **10 ^{−14} m**. Assuming spherical shape, nuclear radii can be calculated according to following formula:

r = r_{0} . A^{1/3}

where r_{0} = 1.2 x 10^{-15 }m = 1.2 fm

If we use this approximation, we therefore expect the **geometrical cross-sections** of nuclei to be of the order of πr^{2} or **4.5×10 ^{−30 }m² for hydrogen** nuclei or

**1.74×10**nuclei.

^{−28}m² for^{238}U## Atomic Mass Number and Nuclear Fission

The nuclear binding energy as a function of the mass number A and the number of

protons Z based on **the liquid drop model** can be written as:

This formula is called **the Weizsaecker Formula** (or **the semi-empirical mass formula**). With the aid of **the Weizsaecker formula** the binding energy can be calculated very well for nearly all isotopes. This formula provides a good fit for heavier nuclei.

From the nuclear binding energy curve and from the table it can be seen that, in the case of splitting a ^{235}U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ^{235}U nucleus. According to the Weizsaecker formula, the total energy released for such reaction will be approximately **235 x (8.5 – 7.6) ≈ 200 MeV.**

As well as the critical energy depends on the nuclear structure and is quite large for light nuclei with Z < 90. For heavier nuclei with Z > 90, the critical energy is about **4 to 6 MeV** for A-even nuclei, and generally is** much lower for A-odd nuclei**.

## Atomic Mass Number and Atomic Mass (Isotopic Mass)

The size and mass of atoms are so small that the use of normal measuring units, while possible, is often inconvenient. Units of measure have been defined for mass and energy on the atomic

scale to make measurements more convenient to express. The unit of measure for mass is the **atomic mass unit (amu)**. One atomic mass unit is equal to 1.66 x 10^{-24} grams.

Besides the standard kilogram, it is a second mass standard. It is the carbon-12 atom, which, by international agreement, has been assigned a mass of 12 atomic mass units (u). The relation between the two units is

One atomic mass unit is equal:

1u = 1.66 x 10^{-24} grams.

One unified atomic mass unit is **approximately** the mass of one nucleon (either a single proton or neutron) and is numerically equivalent to 1 g/mol.

For ^{12}C the atomic mass is exactly 12u, since the atomic mass unit is defined from it. For other isotopes, the isotopic mass usually differs and is usually within 0.1 u of the mass number. For example, ** ^{63}Cu** (29 protons and 34 neutrons) has a mass number of 63 and an isotopic mass in its

**nuclear ground state is 62.91367 u.**

There are two reasons for the difference between mass number and isotopic mass, known as the mass defect:

- The
**neutron is**slightly**heavier**than the**proton**. This increases the mass of nuclei with more neutrons than protons relative to the atomic mass unit scale based on^{12}C with equal numbers of protons and neutrons. - The nuclear binding energy varies between nuclei. A nucleus with greater binding energy has a lower total energy, and therefore a
**lower mass**according to Einstein’s mass-energy equivalence relation*E*=*mc*^{2}. Forthe atomic mass is less than 63 so this must be the dominant factor.^{63}Cu

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