Physics Tutorial: Atomic Nucleus and Its Structural Properties

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In this Physics tutorial, you will learn:

  • What is atom?
  • What are the historical events that led in the discovery of atom?
  • What are the atomic models in a chronological way?
  • What are the Bohr postulates?
  • What is atomic nucleus? What are its components?
  • What is a chemical element? How chemical elements are arranged?
  • What are isotopes and isobars?
  • What is the atomic mass (weight)?
  • How to calculate the dimensions of atomic nuclei?
  • What is the average density of nuclear material?

Introduction

What is matter? What is it made of?

Is matter continuous or discrete? Where do you base your claim?

Do you know what the smallest particle of matter is? Can we divide this particle further? How?

In this article, we will explain some structural properties of atomic nucleus - the hardest and most massive part of atom. In addition, some historical events that brought the important discoveries regarding the atomic structure (albeit not always correct), are briefly discussed, in order to highlight the difficulties that arise from the study of such invisible worlds.

The Atom

Everybody nowadays knows that matter is made up by microscopic particles called atoms. However, this has not always taken as granted. Despite people have doubted since antiquity about the apparent continuous nature of matter (Democritus was the first who used the term "atom" to describe the tiniest particle of matter in the 1st century), only in 19th century scientists widely accepted (albeit without any clear evidence) that matter is not continuous but rather, discrete. They already had realized that matter is composed by atoms, and that it has a well-determined structure in microscopic level that lies beyond the ability of our senses to detect it. However, they had no idea about the atomic structure. The atomic theory of those years produced a unique framework containing all physics and chemistry related phenomena of that period, but without being able to explain a number of mysteries that began to appear towards the end of 19th century. As a result, the old atomic theory resulted as insufficient to describe matter in all its complexity.

New facts discovered during experiments that required a convincing explanation did arise in that period, where the two most significant were:

  1. The discovery of electron in 1887 by Thompson
    In 1887, Thompson discovered the negatively charged electric particles as basic atomic components. These particles were given the name "electrons". This was the first step towards the discovery of the true atomic structure. The fact that electrons contains a negative charge and the entire atom is neutral implied the existence of a balancing positive charge at the same amount inside the atom.
  2. The discovery of penetrating EM radiation or "radioactivity" by Becquerel in 1896
    In 1896, Henry Becquerel was observing the properties of X-rays discovered one year earlier by Rontgen, using natural fluorescent minerals. He believed that if using uranium salt, it would radiate X-rays after absorbing the incident sunlight. For some personal reasons, he gave up from this experiment but anyway, a few days later he developed the photographic films planned for the experiment and surprisingly, he noticed that the images in them were strong and clear without having need for any sunlight exposure. Becquerel therefore realized that uranium radiates naturally, without the need for any external source of energy such as the sunlight. In this way, he unintentionally had discovered the phenomenon of natural radioactivity. He then studied further and more in detail this phenomenon and reached the conclusion that it was not the X-radiation already known but another form of an unknown radiation.
    The inability to explain such phenomena derived from the fact that science at that time had reached to study of atom as a whole only, without being able to go more in detail. For scientists of 19th century, atom represented the last station of knowledge. However, in the light of new facts discovered at that period, it was evident the need for a revised atomic structure, which would give answer to the mysteries of micro-world.

Thompson's Atomic Model (in 1898)

According to the Thomson atomic model, often referred to as the "plum-pudding" model, the atom is a sphere of uniformly distributed positive charge about one angstrom in diameter (1 angstrom = 0.1 nm = 10-10 m). Electrons are embedded in a regular pattern, like raisins in a plum pudding or watermelon, to neutralize the positive charge.

Physics Tutorials: This image provides visual information for the physics tutorial Atomic Nucleus and Its Structural Properties

However, now we know that this model is wrong, as the true atomic structure is completely different. The key discovery in this regard was made in 1911, with the experiment carried out by Rutherford, which we will explain in the next paragraph.

Experiment of Rutherford

Rutherford carried out an experiment for determining the scattering angle of alpha particles (now we know that alpha particles are Helium nuclei but the concept of atomic nuclei was still unknown at that time) when they penetrate a thin golden foil. After emitted from the source, alpha particles were allowed to pass through a very thin hole in order to obtain a regular one-dimensional beam.

After passing through the thin golden foil, most alpha particles continued their motion undisturbed in the original direction. However, a few of them deflected and hit a screen placed at right angle to the golden foil, as shown in the figure.

Physics Tutorials: This image provides visual information for the physics tutorial Atomic Nucleus and Its Structural Properties

The screen was made by zinc sulfide, a material that produces bright spots when hit by alpha particles. The figure shows only one screen but Rutherford placed two parallel screens in both sides of the golden foil in order to convince himself that the phenomenon of bright spots was not casual.

If Thompson model were correct, alpha particles would not deflect as according this model there are many spaces with positive charge inside the atom. Even if any alpha particle encounters an electron during its motion inside the gold atoms, the resulting electric force would be very small to make it deflect so much, i.e. the deflection caused by electric force between alpha particles and electrons is quite unnoticeable (less than one degree).

However, Rutherford noticed that some particles were deflected at a large angle, some even turned back. He was very surprised by this phenomenon. Later, in an interview he declared: "This was incredible - it is like firing a 15-inch artillery shell at a sheet of tissue paper and the shell came back to hit you."

The only reasonable explanation of this strange phenomenon was that the positive charge is not distributed evenly throughout the atom but it is rather concentrated at the centre of atom, where it causes a large repelling force on other positive charges coming towards them (such as the case of alpha particles). Hence, it was clear that the Thompson model of atom is not true and another atomic model that takes into consideration the new discoveries was necessary to introduce.

Rutherford's Atomic Model

In the light of new discoveries, Ernest Rutherford therefore proposed his new version of atomic model. According to this model, an atom consists on a small positively charged nucleus at its centre, where most of atomic mass is concentrated. As for electrons, they revolve around the nucleus to preserve the atom's electric neutrality. Look at the figure:

Physics Tutorials: This image provides visual information for the physics tutorial Atomic Nucleus and Its Structural Properties

This model provides a satisfactory explanation to the phenomenon of alpha particles deflection after penetrating the golden foil. However, in Rutherford's model there is a serious shortcoming. It could not explain the stability of atom in the sense that when electrons revolve around the nucleus, they lose energy and eventually collide with the nucleus.

Hence, despite Rutherford's atomic model was a further step towards the truth, it was incomplete and therefore, it required revision. Niels Bohr made a further advancement in this direction with the new atomic model he proposed.

Bohr's Atomic Model

The issue of atom's stability was resolved by Bohr, who in 1913 proposed a new model, in which electrons move in determined circular orbits around the nucleus, similarly to the revolution of planets around the Sun. This prevents them from losing energy during such revolutions. Physics Tutorials: This image provides visual information for the physics tutorial Atomic Nucleus and Its Structural Properties

With his atomic model, Bohr proposed two courageous postulates:

  1. The postulate of stationary states: An electron can revolve around the nucleus in certain fixed orbits of definite energy without emission of any radiant energy. Such orbits are called stationary orbits.
  2. The postulate of frequencies. An electron can make a transition from a stationary state of higher energy E2 to a state of lower energy E1 and in doing so, it emits a single photon of frequency f, the value of which, is given by
f = E2 - E1/h

where h is the Planck's constant.

A generalized version of the second postulate would be:

"An atom (electron) can emit or absorb radiation only during its transition from one stationary energetic level (orbit) into another."

In a certain sense, the Bohr's atomic model represents the beginning of modern quantum mechanics. Although now this model is outdated, it is still used for convenience to describe the atom, especially in explaining the energetic levels of hydrogen atom.

Now, the next question that arose regarded the structure of atomic nucleus. Is it an object without any internal structure that only manifests some special features or maybe there are other elements inside it? Further investigations and experiments confirmed the second hypothesis, i.e. atomic nucleus contains other particles inside. Let's see this aspect more in detail.

Atomic Nucleus

All data obtained during experiments led in the acceptance of the fact that atomic nucleus is not the smallest thing in the universe; there are other particles inside the atomic nucleus. These particles are protons and neutrons, which have the same mass roughly (neutron is slightly heavier than proton) and both of them are much heavier than electron (proton is 1832 times heavier than electron). More specifically, the mass of a proton is mp = 1.6726 × 10-27 kg and that of neutron is mn = 1.6749 × 10-27 kg. We have seen in Section 16 that protons bear a positive charge while neutrons bear no electric charge (they are neutral).

As explained in other sections, protons and neutrons are often referred to with a common name - nucleons (particles located inside the nucleus). The number of protons in a nucleus (otherwise known as atomic number) is denoted by Z while that of neutrons by N. Hence, the number of nucleons (atomic mass) A is

A = Z + N

(It must be noted here that besides being particles, protons and neutrons like all the other particles manifest wave behavior as well, as explained in the Section 19 of this course.)

The number of protons (atomic number) V indicates the type of element. There are 118 types of (chemical) elements known so far, where each of them has a different atomic number. All of them are arranged in a table of elements, known as the periodic table as shown below.

Physics Tutorials: This image provides visual information for the periodic table

As you see from the periodic table, the name of each chemical element is indicated through abbreviations. For example, hydrogen is indicated by H, helium by He, lithium by Li, and so on. Symbolically (especially in nuclear reactions), the number of protons Z is written as a subscript before any element X (on its left) while the number of nucleons N is written as a superscript before the element symbol. Hence, the general form of nucleus of a chemical element is

AZX

For example, a helium nucleus that has two protons and two neutrons is written as

42E

Sometimes, the number of nucleons is written following the element. For example, the symbol C-12 means the given carbon element contains 12 nucleons (in which 6 are protons and 6 neutrons).

Some elements are stable, other are less stable. In general, higher the atomic number Z, less stable the element is. For example, the main two elements in the Sun are hydrogen H and helium He, which have 1 and 2 protons in their nuclei respectively. All the other elements have disappeared due to strong explosions frequently occurring in the Sun.

All elements heavier than Uranium (Z = 92) are human made, i.e. they are obtained through artificial methods and are therefore unstable, as they cannot exist naturally.

Isotopes and Isobars

If the number of protons Z in two different atoms is equal but the number of neutrons differs, these atoms are known as isotopes. However, both atoms still represent the same chemical element despite having a different atomic mass. For example, the isotopes of Lithium, Li range from A = 3 to A = 12. This means a lithium atom may contain from 0 to 9 neutrons as Z (He) = 3. Isotopes have slight variations in their physical and chemical behavior.

On the other hand, if two different chemical elements have the same number of nucleons A, they are called isobars. Obviously, isobars differ from each other much more than isotopes, as isobars represent different materials, despite the atomic mass is almost equal. For example, S-40, Cl-40, Ar-40, K-40, and Ca-40 are all isobars because all of them contain 40 particles in their respective nuclei. However, the difference in naming them indicates that they belong to different elements. Thus, from the periodic table shown above, we can see that the atomic numbers Z of these elements are 16, 17, 18, 19 and 20 respectively.

Atomic Mass (Weight) Number

Another number that appear near each element (below its symbol) in the periodic table is the atomic mass or atomic weight number A. In general, it is not a whole number and shows a mean value of all isotopes of a given element based on the percentage of their existence. However, the value of A is easily found by rounding the value shown in the periodic table (under the name of element) to the nearest whole number. Atomic weight is not used to identify the name of element (Z is enough for this) but the corresponding isotope instead. Hence, Z = 1 and A = 3 means we are dealing with a hydrogen isotope (known as deuterium) which has 1 proton and 2 neutrons in the nucleus.

The unit used to indicate the atomic mass is not kilogram but amu (atomic mass unit) or simply u instead. Thus, in scientific terms, 1 u represents 1/12 of the mass of a C-12 nucleus and it is a rough value to indicate the mass of one proton or one neutron. The conversion factor between amu and kilogram is

1 amu = 1.6605 × 10-27 kg

For example, in the periodic table shown above, you see the number 195.085 below the Platinium (Pt) element (Z = 78). This means a Platinium nucleus is about 195 times heavier than 1/12 of a C-12 nucleus. Another conclusion we draw from the above value is that A (Pt) = 195, so the number of neutrons in this nucleus is N = A - Z = 195 - 78 = 118.

Example 1

A chemical element X has 26 protons and 57 particles in the nucleus.

  1. What element is it?
  2. What is the number of neutrons in the nuclei of X?
  3. What is the atomic weight of this element?

Solution 1

  1. The atomic number Z of the unknown element must be 26. Therefore, we identify this element in the periodic table. It is shown below. Physics Tutorials: This image provides visual information for the physics tutorial Atomic Nucleus and Its Structural Properties Obviously, the name of element is iron (its symbol is Fe).
  2. Since the element has 57 particles in the nucleus (A = 57), the number of neutrons N is
    A = Z + N
    N = A - Z
    = 57 - 28
    = 29
    Thus, the given element is one of many isotopes of iron.
  3. The figure extracted from the periodic table shows the value 55.845 below the name of element. This shows that the atomic weight (mass) of this element is 55.845 amu.

Dimensions of Atomic Nucleus

The nucleus of an atom has very small dimensions, much smaller compared to those of the atom itself. Experiments show that if the nucleus is thought as a sphere, its radius ranges from 10-14 m to 10-15 m. The empirical formula

r ≈ r0 ∙ A-1/3 [metres]

where r0 is a constant (r0 = 1.2 × 10-15 m), is often used to calculate the radius r of nucleus for any element, and A is the number of nucleons contained in the nucleus of the given element. Thus, the formula that calculates the radius of atomic nuclei is

r ≈ 1.2 × 10-15 ∙ A1/3 [metres]

Example 2

Calculate the radius of a hydrogen isotope nuclei known as deuterium (Z = 1) if it contains one neutron inside.

Solution 2

Deuterium atoms contain A = Z + N = 1 + 1 = 2 particles in their nuclei. Thus, the radius r of deuterium nucleus is

r ≈ r0 ∙ A1/3
r ≈ 1.2 × 10-15 ∙ A1/3
= 1.2 × 10-15 ∙ 21/3 m
≈ 1.51 × 10-15 m

When compared to the dimensions of the atom ( ≈ 10-10 m), this value obtained for the radius of hydrogen nuclei is about 105 times smaller (about 100 000 times smaller) than the radius of atom itself. Thus, a nucleus in an atom is comparable to a rice grain (about 1 mm in thickness) placed at centre of a football field (about 100 m long).

Average Density of "Nuclear Material"

Despite the volume of atomic nucleus is much smaller compared to the volume of the atom itself, the mass of atom is almost entirely concentrated in the nucleus (only electrons that are much lighter than protons and neutrons are outside the nucleus). Therefore, the density of atomic nuclei is much greater than the density of the corresponding atoms. Let's clarify this fact through an example.

Example 3

Calculate the average density of nuclear material in a nucleus of atomic mass A.

Solution 3

Assuming the shape of atomic nuclei as spheres, we have for the volume of nuclei in terms of atomic mass A:

V = 4/3 π ∙ r3
= 4/3 π ∙ (r0 ∙ A1/3 )3
= 4/3 π ∙ (1.2 × 10-15 ∙ A1/3 )3
= 7.235 × 10-45 ∙ A [m3]

The density of nuclear material expressed in nucleons per cubic metre is

ρ = A/V
= A/7.235 × 10-45 ∙ A
= 1/7.235 × 10-45 m3
= 1.38 × 1044 nucleons/m3

Since the mass of a nucleon (1 amu therefore) is about 1.6605 × 10-27 kg, we obtain for the density of atomic nuclei in kg/m3:

ρ = 1.38 × 1044 nucleons/m3 × 1.6605 × 10-27 kg/nucleon
= 2.29 × 1017 kg/m3

This value is very large; it is about 2.29 × 1014 times the density of water (103 kg/m3). Another thing to point out here is that we have not specified the type of nuclei in this example. This means that nuclear density is more or less the same for all types of materials. Such high densities are found only in "packed" nuclei that exist only in in heavy stars. In these stars, atoms have lost their electrons due to high temperatures and as a result, the nuclei are much closer to each other than in normal atoms. The average density of such stars ranges from 107 kg/m3 to 1010 kg/m3.

Summary

Nowadays, everybody knows that matter is made up by microscopic particles called atoms. However, the atomic structure was discovered only about one century ago, despite people have been aware of particle nature of matter since antiquity.

In 1887, Thompson discovered the negatively charged electric particles as basic atomic components. These particles were given the name "electrons". The fact that electrons contains a negative charge and the entire atom is neutral implied the existence of a balancing positive charge at the same amount inside the atom.

In 1888, Becquerel realized that uranium radiates naturally, without the need for any external source of energy such as the sunlight. In this way, he unintentionally discovered the phenomenon of natural radioactivity. He then studied further and more in detail this phenomenon and reached the conclusion that it was not the X-radiation already known but another form of an unknown radiation.

According to the Thomson atomic model (1898), often referred to as the "plum-pudding" model, the atom is a sphere of uniformly distributed positive charge about one angstrom in diameter (1 angstrom = 0.1 nm = 10-10 m). Electrons are embedded in a regular pattern, like raisins in a plum pudding or watermelon, to neutralize the positive charge.

The key discovery in this regard was made in 1911, with the experiment carried out by Rutherford. According to his model, an atom consists on a small positively charged nucleus at its centre, where most of atomic mass is concentrated. As for electrons, they revolve around the nucleus to preserve the atom's electric neutrality. However, this model could not explain the stability of atom in the sense that when electrons revolve around the nucleus, they lose energy and eventually collide with the nucleus.

The issue of atom's stability was resolved by Bohr, who in 1913 proposed a new model, in which electrons move in determined circular orbits around the nucleus, similarly to the revolution of planets around the Sun. This prevents them from losing energy during such revolutions.

With his atomic model, Bohr proposed two courageous postulates:

  1. The postulate of stationary states: An electron can revolve around the nucleus in certain fixed orbits of definite energy without emission of any radiant energy. Such orbits are called stationary orbits.
  2. The postulate of frequencies. An electron can make a transition from a stationary state of higher energy E2 to a state of lower energy E1 and in doing so, it emits a single photon of frequency f.

A generalized version of the second postulate would be:

"An atom (electron) can emit or absorb radiation only during its transition from one stationary energetic level (orbit) into another."

Atomic nucleus is not the smallest thing in the universe; there are other particles inside it. These particles are protons and neutrons. Protons and neutrons are often referred to with a common name - nucleons (particles located inside the nucleus). The number of protons in a nucleus (otherwise known as atomic number) is denoted by Z while that of neutrons by N. Hence, the number of nucleons (atomic mass) A is

A = Z + N

The number of protons (atomic number) Z indicates the type of element. There are 118 types of (chemical) elements known so far, where each of them has a different atomic number. All of them are arranged in a table of elements, known as the periodic table.

If the number of protons Z in two different atoms is equal but the number of neutrons differs, these atoms are known as isotopes. However, both atoms still represent the same chemical element despite having a different atomic mass.

On the other hand, if two different chemical elements have the same number of nucleons A, they are called isobars. They differ from each other much more than isotopes, as isobars represent different materials, despite the atomic mass is almost equal.

Another number that appear near each element (below its symbol) in the periodic table is the atomic mass or atomic weight number A. In general, it is not a whole number and shows a mean value of all isotopes of a given element based on the percentage of their existence.

The unit used to indicate the atomic mass is not kilogram but amu (atomic mass unit) or simply u instead. Thus, in scientific terms, 1 u represents 1/12 of the mass of a C-12 nucleus and it is a rough value to indicate the mass of one proton or one neutron. The conversion factor between amu and kilogram is

1 amu = 1.6605 × 10-27 kg

The nucleus of an atom has very small dimensions, much smaller compared to those of the atom itself. Experiments show that if the nucleus is thought as a sphere, its radius ranges from 10-14 m to 10-15 m. The empirical formula

r ≈ r0 ∙ A-1/3 [metres]

where r0 is a constant (r0 = 1.2 × 10-15 m), is often used to calculate the radius r of nucleus for any element, and A is the number of nucleons contained in the nucleus of the given element. Thus, the formula that calculates the radius of atomic nuclei is

r ≈ 1.2 × 10-15 ∙ A1/3 [metres]

Despite the volume of atomic nucleus is much smaller compared to the volume of the atom itself, the mass of atom is almost entirely concentrated in the nucleus (only electrons that are much lighter than protons and neutrons are outside the nucleus). Therefore, the density of atomic nuclei is much greater than the density of the corresponding atoms.

Atomic Nucleus and Its Structural Properties Revision Questions

1. What is the excess of neutrons in the Cu-64 chemical element?

  1. 0
  2. 29
  3. 35
  4. 64

Correct Answer: C

2. Calculate the radius of 4420Ca nuclei in metres.

  1. 4.236 × 10-15 m
  2. 3.530 × 10-15 m
  3. 1.200 × 10-15 m
  4. 7.960 × 10-15 m

Correct Answer: A

3. What would the radii of 23892U nuclei be if their density were the same as the density of Earth (about 5500 kg/m3)?

  1. 7.185 × 10-21 m
  2. 1.197 × 10-7 m
  3. 3.952 × 10-25 m
  4. 2.58 × 10-10 m

Correct Answer: D

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