Properties of Nuclei

Samuel J. Ling; Jeff Sanny; William Moebs

Nuclear Physics

Properties of Nuclei

Samuel J. Ling; Jeff Sanny; and William Moebs

Learning Objectives

By the end of this section, you will be able to:

Describe the composition and size of an atomic nucleus
Use a nuclear symbol to express the composition of an atomic nucleus
Explain why the number of neutrons is greater than protons in heavy nuclei
Calculate the atomic mass of an element given its isotopes

The atomic nucleus is composed of protons and neutrons ((Figure)). Protons and neutrons have approximately the same mass, but protons carry one unit of positive charge $\left(\text{+}e\right),$ and neutrons carry no charge. These particles are packed together into an extremely small space at the center of an atom. According to scattering experiments, the nucleus is spherical or ellipsoidal in shape, and about 1/100,000th the size of a hydrogen atom. If an atom were the size of a major league baseball stadium, the nucleus would be roughly the size of the baseball. Protons and neutrons within the nucleus are called nucleons.

The atomic nucleus is composed of protons and neutrons. Protons are shown in blue, and neutrons are shown in red.

The figure shows a cluster of red and blue spheres packed closely together. The red spheres are labeled neutrons and the blue ones protons.

Counts of Nucleons

The number of protons in the nucleus is given by the atomic number, Z. The number of neutrons in the nucleus is the neutron number, N. The total number of nucleons is the mass number, A. These numbers are related by

$A=Z+N.$

A nucleus is represented symbolically by

${}_{Z}^{A}\text{X},$

where X represents the chemical element, A is the mass number, and Z is the atomic number. For example, ${}_{\phantom{\rule{0.5em}{0ex}}6}^{12}\text{C}$ represents the carbon nucleus with six protons and six neutrons (or 12 nucleons).

A graph of the number N of neutrons versus the number Z of protons for a range of stable nuclei (nuclides) is shown in (Figure). For a given value of Z, multiple values of N (blue points) are possible. For small values of Z, the number of neutrons equals the number of protons $\left(N=P\right),$ and the data fall on the red line. For large values of Z, the number of neutrons is greater than the number of protons $\left(N>P\right),$ and the data points fall above the red line. The number of neutrons is generally greater than the number of protons for $Z>15.$

This graph plots the number of neutrons N against the number of protons Z for stable atomic nuclei. Larger nuclei, have more neutrons than protons.

A graph showing number of neutrons, N versus number of protons, Z. A straight line on the graph is labeled N equal to Z. Another, jagged line, is labeled band of stability. This has incremental steps. It starts at the origin. At Z = 80, the value of N is 120.

A chart based on this graph that provides more detailed information about each nucleus is given in (Figure). This chart is called a chart of the nuclides. Each cell or tile represents a separate nucleus. The nuclei are arranged in order of ascending Z (along the horizontal direction) and ascending N (along the vertical direction).

Partial chart of the nuclides. For stable nuclei (dark blue backgrounds), cell values represent the percentage of nuclei found on Earth with the same atomic number (percent abundance). For the unstable nuclei, the number represents the half-life.

Figure shows a chart of nuclides, with ascending Z along the horizontal direction and ascending N along the vertical direction. The cells along diagonal in the centre of the chart are color coded to indicate that they are stable.

Atoms that contain nuclei with the same number of protons (Z) and different numbers of neutrons (N) are called isotopes. For example, hydrogen has three isotopes: normal hydrogen (1 proton, no neutrons), deuterium (one proton and one neutron), and tritium (one proton and two neutrons). Isotopes of a given atom share the same chemical properties, since these properties are determined by interactions between the outer electrons of the atom, and not the nucleons. For example, water that contains deuterium rather than hydrogen (“heavy water”) looks and tastes like normal water. The following table shows a list of common isotopes.

Common Isotopes*No entry if less than 0.001 (trace amount).
**Stable if half-life > 10 seconds.
Element	Symbol	Mass Number	Mass (Atomic Mass Units)	Percent Abundance*	Half-life**
Hydrogen	H	1	1.0078	99.99	stable
	${}^{2}\text{H}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\text{D}$	2	2.0141	0.01	stable
	${}^{3}\text{H}$	3	3.0160	−	12.32 y
Carbon	${}^{12}\text{C}$	12	12.0000	98.91	stable
	${}^{13}\text{C}$	13	13.0034	1.1	stable
	${}^{14}\text{C}$	14	14.0032	−	5730 y
Nitrogen	${}^{14}\text{N}$	14	14.0031	99.6	stable
	${}^{15}\text{N}$	15	15.0001	0.4	stable
	${}^{16}\text{N}$	16	16.0061	−	7.13 s
Oxygen	${}^{16}\text{O}$	16	15.9949	99.76	stable
	${}^{17}\text{O}$	17	16.9991	0.04	stable
	${}^{18}\text{O}$	18	17.9992	0.20	stable
	${}^{19}\text{O}$	19	19.0035	−	26.46 s

Why do neutrons outnumber protons in heavier nuclei ((Figure))? The answer to this question requires an understanding of forces inside the nucleus. Two types of forces exist: (1) the long-range electrostatic (Coulomb) force that makes the positively charged protons repel one another; and (2) the short-range strong nuclear force that makes all nucleons in the nucleus attract one another. You may also have heard of a “weak” nuclear force. This force is responsible for some nuclear decays, but as the name implies, it does not play a role in stabilizing the nucleus against the strong Coulomb repulsion it experiences. We discuss strong nuclear force in more detail in the next chapter when we cover particle physics. Nuclear stability occurs when the attractive forces between nucleons compensate for the repulsive, long-range electrostatic forces between all protons in the nucleus. For heavy nuclei $\left(Z>15\right),$ excess neutrons are necessary to keep the electrostatic interactions from breaking the nucleus apart, as shown in (Figure).

(a) The electrostatic force is repulsive and has long range. The arrows represent outward forces on protons (in blue) at the nuclear surface by a proton (also in blue) at the center. (b) The strong nuclear force acts between neighboring nucleons. The arrows represent attractive forces exerted by a neutron (in red) on its nearest neighbors.

Figure a shows a cluster of small red and blue circles. There is a blue proton in the center, surrounded by red neutrons. There are more protons at the periphery, which have arrows pointing outwards. Figure b shows the same cluster. Arrows show both protons and neutrons being attracted towards an adjacent neutron.

Because of the existence of stable isotopes, we must take special care when quoting the mass of an element. For example, Copper (Cu) has two stable isotopes:

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{}_{29}^{63}\text{C}\text{u}\phantom{\rule{0.2em}{0ex}}\left(62.929595\phantom{\rule{0.2em}{0ex}}\text{g/mol}\right)\phantom{\rule{0.2em}{0ex}}\text{with an abundance of}\phantom{\rule{0.2em}{0ex}}69.09\text{%}

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{}_{29}^{65}\text{C}\text{u}\phantom{\rule{0.2em}{0ex}}\left(64.927786\phantom{\rule{0.2em}{0ex}}\text{g/mol}\right)\phantom{\rule{0.2em}{0ex}}\text{with an abundance of}\phantom{\rule{0.2em}{0ex}}30.91\text{%}

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Given these two “versions” of Cu, what is the mass of this element? The atomic mass of an element is defined as the weighted average of the masses of its isotopes. Thus, the atomic mass of Cu is ${m}_{\text{Cu}}=\left(62.929595\right)\left(0.6909\right)\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}\left(64.927786\right)\left(0.3091\right)=63.55\phantom{\rule{0.2em}{0ex}}\text{g/mol}.$ The mass of an individual nucleus is often expressed in atomic mass units (u), where $\text{u}=1.66054\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\text{kg}$ . (An atomic mass unit is defined as 1/12th the mass of a ${}^{12}\text{C}$ nucleus.) In atomic mass units, the mass of a helium nucleus (A = 4) is approximately 4 u. A helium nucleus is also called an alpha (α) particle.

Nuclear Size

The simplest model of the nucleus is a densely packed sphere of nucleons. The volume V of the nucleus is therefore proportional to the number of nucleons A, expressed by

$V=\frac{4}{3}\phantom{\rule{0.2em}{0ex}}\pi {r}^{3}=kA,$

where r is the radius of a nucleus and k is a constant with units of volume. Solving for r, we have

$r={r}_{0}{A}^{1\text{/}3}$

where ${r}_{0}$ is a constant. For hydrogen ( $A=1$ ), ${r}_{0}$ corresponds to the radius of a single proton. Scattering experiments support this general relationship for a wide range of nuclei, and they imply that neutrons have approximately the same radius as protons. The experimentally measured value for ${r}_{0}$ is approximately 1.2 femtometer (recall that $1\phantom{\rule{0.2em}{0ex}}\text{fm}={10}^{-15}\text{m}$ ).

The Iron NucleusFind the radius (r) and approximate density ( $\rho$ ) of a Fe-56 nucleus. Assume the mass of the Fe-56 nucleus is approximately 56 u.

Strategy (a) Finding the radius of ${}^{56}\text{Fe}$ is a straightforward application of $r={r}_{0}{A}^{1\text{/}3}$ , given $A=56.$ (b) To find the approximate density of this nucleus, assume the nucleus is spherical. Calculate its volume using the radius found in part (a), and then find its density from $\rho =m\text{/}V.$

Solution

The radius of a nucleus is given by

$r={r}_{0}{A}^{1\text{/}3}.$

Substituting the values for ${r}_{0}$ and A yields

$\begin{array}{cc}\hfill r& =\left(1.2\phantom{\rule{0.2em}{0ex}}\text{fm}\right){\left(56\right)}^{1\text{/}3}=\left(1.2\phantom{\rule{0.2em}{0ex}}\text{fm}\right)\left(3.83\right)\hfill \\ & =4.6\phantom{\rule{0.2em}{0ex}}\text{fm}.\hfill \end{array}$
Density is defined to be $\rho =m\text{/}V,$ which for a sphere of radius r is

$\rho =\frac{m}{V}=\frac{m}{\left(4\text{/}3\right)\text{π}{r}^{3}}.$

Substituting known values gives

$\rho =\frac{56\phantom{\rule{0.2em}{0ex}}\text{u}}{\left(1.33\right)\left(3.14\right){\left(4.6\phantom{\rule{0.2em}{0ex}}\text{fm}\right)}^{3}}=0.138\phantom{\rule{0.2em}{0ex}}{\text{u/fm}}^{3}.$

Converting to units of $\text{kg}\text{/}{\text{m}}^{3},$ we find

$\rho =\left(0.138\phantom{\rule{0.2em}{0ex}}{\text{u/fm}}^{3}\right)\left(1.66\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\phantom{\rule{0.2em}{0ex}}\text{kg/u}\right)\left(\frac{1\phantom{\rule{0.2em}{0ex}}\text{fm}}{{10}^{-15}\phantom{\rule{0.2em}{0ex}}\text{m}}\right)=2.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{17}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}.$

Significance

The radius of the Fe-56 nucleus is found to be approximately 5 fm, so its diameter is about 10 fm, or ${10}^{-14}\text{m}.$ In previous discussions of Rutherford’s scattering experiments, a light nucleus was estimated to be ${10}^{-15}\text{m}$ in diameter. Therefore, the result shown for a mid-sized nucleus is reasonable.
The density found here may seem incredible. However, it is consistent with earlier comments about the nucleus containing nearly all of the mass of the atom in a tiny region of space. One cubic meter of nuclear matter has the same mass as a cube of water 61 km on each side.

Check Your Understanding Nucleus X is two times larger than nucleus Y. What is the ratio of their atomic masses?

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Summary

The atomic nucleus is composed of protons and neutrons.
The number of protons in the nucleus is given by the atomic number, Z. The number of neutrons in the nucleus is the neutron number, N. The number of nucleons is mass number, A.
Atomic nuclei with the same atomic number, Z, but different neutron numbers, N, are isotopes of the same element.
The atomic mass of an element is the weighted average of the masses of its isotopes.

Conceptual Questions

Define and make clear distinctions between the terms neutron, nucleon, nucleus, and nuclide.

The nucleus of an atom is made of one or more nucleons. A nucleon refers to either a proton or neutron. A nuclide is a stable nucleus.

What are isotopes? Why do isotopes of the same atom share the same chemical properties?

Problems

Find the atomic numbers, mass numbers, and neutron numbers for (a) ${}_{29}^{58}\text{C}\text{u},$ (b) ${}_{11}^{24}\text{N}\text{a},$ (c) ${}_{\phantom{\rule{0.5em}{0ex}}84}^{210}\text{P}\text{o},$ (d) ${}_{20}^{45}\text{C}\text{a},$ and (e) ${}_{\phantom{\rule{0.5em}{0ex}}82}^{206}\text{P}\text{b}$ .

Use the rule $A=Z+N.$

	Atomic Number (Z)	Neutron Number (N)	Mass Number (A)
(a)	29	29	58
(b)	11	13	24
(c)	84	126	210
(d)	20	25	45
(e)	82	124	206

Silver has two stable isotopes. The nucleus, ${}_{\phantom{\rule{0.5em}{0ex}}47}^{107}\text{A}\text{g},$ has atomic mass 106.905095 g/mol with an abundance of

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; whereas ${}_{\phantom{\rule{0.5em}{0ex}}47}^{109}\text{A}\text{g}$ has atomic mass 108.904754 g/mol with an abundance of

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48.17\text{%}

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. Find the atomic mass of the element silver.

The mass (M) and the radius (r) of a nucleus can be expressed in terms of the mass number, A. (a) Show that the density of a nucleus is independent of A. (b) Calculate the density of a gold (Au) nucleus. Compare your answer to that for iron (Fe).

a. $r={r}_{0}{A}^{1\text{/}3},\rho =\frac{3\phantom{\rule{0.2em}{0ex}}\text{u}}{4\pi {r}_{0}{}^{3}}$ ;
b. $\begin{array}{ccc}\hfill \rho & =\hfill & 2.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{17}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}\hfill \end{array}$

A particle has a mass equal to 10 u. If this mass is converted completely into energy, how much energy is released? Express your answer in mega-electron volts (MeV). (Recall that $1\phantom{\rule{0.2em}{0ex}}\text{eV}=1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\text{J}$ .)

Find the length of a side of a cube having a mass of 1.0 kg and the density of nuclear matter.

side length $=1.6\phantom{\rule{0.2em}{0ex}}\text{μm}$

The detail that you can observe using a probe is limited by its wavelength. Calculate the energy of a particle that has a wavelength of $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-16}\text{m}$ , small enough to detect details about one-tenth the size of a nucleon.

Glossary

atomic mass: total mass of the protons, neutrons, and electrons in a single atom

atomic mass unit: unit used to express the mass of an individual nucleus, where $1\text{u}=1.66054\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\phantom{\rule{0.2em}{0ex}}\text{kg}$

atomic nucleus: tightly packed group of nucleons at the center of an atom

atomic number: number of protons in a nucleus

chart of the nuclides: graph comprising stable and unstable nuclei

isotopes: nuclei having the same number of protons but different numbers of neutrons

mass number: number of nucleons in a nucleus

neutron number: number of neutrons in a nucleus

nucleons: protons and neutrons found inside the nucleus of an atom

nuclide: nucleus

radius of a nucleus: radius of a nucleus is defined as $r={r}_{0}{A}^{1\text{/}3}$

strong nuclear force: force that binds nucleons together in the nucleus

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