Material Properties

Geiger-Marsden Experiments – Rutherford gold foil experiment

The Rutherford model of the atom is a model of the atom devised by the British physicist Ernest Rutherford. Rutherford’s new model for the atom is based on the experimental results, which were obtained from Geiger-Marsden experiments (also called the Rutherford gold foil experiment). The Geiger–Marsden experiments were performed between 1908 and 1913 by Hans Geiger (of Geiger counter fame) and Ernest Marsden (a 20-year-old student who had not yet earned his bachelor’s degree) under the direction of Ernest Rutherford.

Rutherford’s idea was to direct energetic alpha particles at a thin metal foil and measure how an alpha particle beam is scattered when it strikes a thin metal foil. A narrow collimated beam of alpha particles was aimed at a gold foil of approximately 1 μm thickness (about 10,000 atoms thick).  Alpha particles are energetic nuclei of helium (usually about 6 MeV). Alpha particles, which are about 7300 times more massive than electrons, have a positive charge of +2e. Because of their relatively much greater mass, alpha particles are not significantly deflected from their paths by the electrons in the metal’s atoms.

According to Thomson’s model, if an alpha particle were to collide with an plum-pudding atom, it would just fly straight through, its path being deflected by at most a fraction of a degree. But in the experiment Geiger and Marsden saw that most of the particles are scattered through rather small angles, but, and this was the big surprise, a very small fraction of them are scattered through very large angles, approaching 180° (i.e., they were recoiled backwards).

In Rutherford’s words:

“It is almost as incredible as if you had fired a fifteen inch shell at a sheet of tissue paper and it came back and hit you.”

Rutherford assumed that, to deflect the alpha particle backward, there must be a very large force. This force could be provided only result from a collision with a massive target or from an interaction with an electric or magnetic field of great strength. In previous experiments, it was shown the deflections had to be electrical or perhaps possibly magnetic in origin.

As a result, Rutherford abandoned the Thomson model. Geiger-Marsden experiments were therefore a landmark series of experiments by which scientists discovered that every atom contains a nucleus (whose diameter is of the order 10^-14m.) where all of its positive charge and most of its mass are concentrated in a small region called an atomic nucleus. In Rutherford’s atom, the diameter of its sphere (about 10^-10 m) of influence is determined by its electrons. In other words, the nucleus occupies only about 10^-12 of the total volume of the atom or less (the nuclear atom is largely empty space), but it contains all the positive charge and at least 99.95% of the total mass of the atom.

Based on these results, Ernest Rutherford proposed a new model of the atom. He postulated that the positive charge in an atom is concentrated in a small region (in comparison to the rest of the atom) called a nucleus at the center of the atom with electrons existing in orbits around it. Furthermore, the nucleus is responsible for most of the mass of the atom.

See also:

Atomic Theory

Discovery of Atomic Nucleus

In physics, the atomic nucleus is the central part of an atom. In comparison to an atom, it is much more smaller and contains most of the mass of the atom. The atomic nucleus also contains all of its positive electric charge (in protons), while all of its negative charge is distributed in the electron cloud.

The atomic nucleus was discovered by Ernest Rutherford, who proposed a new model of the atom based on Geiger-Marsden experiments. These experiments were performed between 1908 and 1913 by Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford. Rutherford’s idea was to direct energetic alpha particles at a thin metal foil and measure how an alpha particle beam is scattered when it strikes a thin metal foil. A narrow collimated beam of alpha particles was aimed at a gold foil of approximately 1 μm thickness (about 10,000 atoms thick).  Alpha particles are energetic nuclei of helium (usually about 6 MeV). Alpha particles, which are about 7300 times more massive than electrons, have a positive charge of +2e. Because of their relatively much greater mass, alpha particles are not significantly deflected from their paths by the electrons in the metal’s atoms.

According to Thomson’s model, if an alpha particle were to collide with an plum-pudding atom, it would just fly straight through, its path being deflected by at most a fraction of a degree. But in the experiment Geiger and Marsden saw that most of the particles are scattered through rather small angles, but, and this was the big surprise, a very small fraction of them are scattered through very large angles, approaching 180° (i.e., they were recoiled backwards).

In Rutherford’s words:

“It is almost as incredible as if you had fired a fifteen inch shell at a sheet of tissue paper and it came back and hit you.”

Rutherford assumed that, to deflect the alpha particle backward, there must be a very large force. This force could be provided only result from a collision with a massive target or from an interaction with an electric or magnetic field of great strength. In previous experiments, it was shown the deflections had to be electrical or perhaps possibly magnetic in origin.

These experiments were a landmark series of experiments by which scientists discovered that every atom contains a nucleus (whose diameter is of the order 10^-14m) where all of its positive charge and most of its mass are concentrated in a small region called an atomic nucleus. In Rutherford’s atom, the diameter of its sphere (about 10^-10 m) of influence is determined by its electrons. In other words, the nucleus occupies only about 10^-12 of the total volume of the atom or less (the nuclear atom is largely empty space), but it contains all the positive charge and at least 99.95% of the total mass of the atom.

Based on these results, Ernest Rutherford proposed a new model of the atom. He postulated that the positive charge in an atom is concentrated in a small region (in comparison to the rest of the atom) called a nucleus at the center of the atom with electrons existing in orbits around it. Furthermore, the nucleus is responsible for most of the mass of the atom.

See also:

Atomic Theory

Saturnian Model of the Atom – Nagaoka’s Planetary Model

It must be noted, the Rutherford model of the atom was not the first model, which proposed a nuclear structure. The idea of a nuclear structure was previously proposed in 1903 by a Japanese physicist Hantaro Nagaoka, who rejected Thomson’s model on the grounds that opposite charges are impenetrable.His model was known as the Saturnian model of the atom and according to this model the atom consists of a massive positive centre surrounded by a number of orbiting electrons, in the manner of Saturn and its rings.

Nagaoka’s planetary model had two predictions:

• a very massive atomic center (in analogy to a very massive planet)
• electrons revolving around the nucleus, bound by electrostatic forces

Both predictions were successfully confirmed by Ernest Rutherford in 1911, however, other details of the model was incorrect. Notwithstanding the Rutherford model of the atom was really very close to modern concept of the atom, it was based on classical physics. But neither classical physics nor the Rutherford model can explain the following problem. Why the negatively charged electrons are prevented from falling into the positively charged nucleus? Forces between static electrically charged particles are governed by the Coulomb’s law:

Rutherford suggested that perhaps the electrons revolve in orbits about the nucleus. In classical mechanics, a condition for the dynamic stability of such systems of rotating particles is that only forces of attraction act between its components.

Classical physics also states that any accelerating electric charge (either oscillating or revolving) emits electromagnetic radiation. As a result, an electron orbiting inside an atom should be continually losing energy and it should spiral into the nucleus within a fraction of a second. In fact, physicists calculated that the electron should lose all of its energy and spiral down into the proton in only about 0.000000000001 second, which means that every atoms should not exist longer than a mere 10^-12 seconds.

The solution to the problem of the stability of the electrons in an atom was provided by young Danish physicist Niels Bohr working with Ernest Rutherford at the University of Manchester in 1913. His model adopted Planck’s quantum hypothesis and he proposed a model in which the electrons of an atom were assumed to orbit the nucleus but could only do so in a finite set of orbits. He postulated that an atom emits or absorbs energy only in discrete quanta corresponding to absorption or radiation of a photon.

See also:

Atomic Theory

Discovery of Proton

Before the discovery of atomic nucleus, there was ideas that all atoms are composed of hydrogen atoms (called by William Prout “protyles“). This hypothesis is known as Prout’s hypothesis. In 1816, he hypothesized that the hydrogen atom was the only truly fundamental particles, and that the other atoms were actually groupings of various numbers of hydrogen atoms (protyles). Further important facts were discovered in 1886 by German physicist Eugen Goldstein. He discovered anode rays and showed that they were positively charged particles (ions) produced from gases. In 1907 a study of how these rays were deflected in a magnetic field, revealed that the particles making up the ray were not all the same mass. The lightest ones, formed when there was some hydrogen gas in the tube, were calculated to be about 1840 times as massive as an electron.

Following Geiger-Marsden experiments, which were a landmark series of experiments by which scientists discovered that every atom contains an atomic nucleus (whose diameter is of the order 10^-14m) lead to the identification of the ‘primitive atom’ . The identification of the primitive atom of positive electricity was first made by the British physicist Ernest Rutherford. Rutherford proved that the hydrogen nucleus is present in other nuclei, a result usually described as the discovery of protons. No positively charged atom lighter than these had ever been detected. Furthermore, they never carry more than a single natural unit of positive charge, +e.

According to modern physics, a proton is one of the subatomic particles that make up matter. In the universe, protons are abundant, making up about half of all visible matter. It has a positive electric charge (+1e) and a rest mass equal to 1.67262 × 10⁻²⁷ kg (938.272 MeV/c²)— marginally lighter than that of the neutron but nearly 1836 times greater than that of the electron. The proton has a mean square radius of about 0.87 × 10⁻¹⁵ m, or 0.87 fm, and it is a spin – ½ fermion.

The protons exist in the nuclei of typical atoms, along with their neutral counterparts, the neutrons. Neutrons and protons, commonly called nucleons, are bound together in the atomic nucleus, where they account for 99.9 percent of the atom’s mass. Research in high-energy particle physics in the 20th century revealed that neither the neutron nor the proton is not the smallest building block of matter.

See also:

Atomic Theory

Bohr Model of the Atom

In atomic physics, the Bohr model if the atom (also known as the Rutherford-Bohr model) is modern model of the hydrogen atom introduced by Danish physicist Niels Bohr working with Ernest Rutherford at the University of Manchester in 1913.

This model provides especially the solution to the problem of the failure of classical physics in the field of atomic physics. Classical electromagnetic theory, makes three entirely wrong predictions about atoms:

• atoms should emit light continuously,
• atoms should be unstable,
• the light they emit should have a continuous spectrum.

The Bohr model adopted Planck’s quantum hypothesis and he proposed a model in which the electrons of an atom were assumed to orbit the nucleus but could only do so in a finite set of orbits. Planck’s quantum hypothesis (Planck’s law) is named after a German theoretical physicist Max Planck, who proposed it in 1900. Planck’s law is a pioneering result of modern physics and quantum theory. Planck’s hypothesis that energy is radiated and absorbed in discrete “quanta” (or energy packets) precisely matched the observed patterns of blackbody radiation and resolved the ultraviolet catastrophe.

Based on this hypothesis, Bohr postulated that an atom emits or absorbs energy only in discrete quanta corresponding to absorption or radiation of a photon. During the emission of a photon, the internal energy of the atom changes by an amount equal to the energy of the photon. Therefore, each atom must be able to exist with only certain specific values of internal energy. Each of these stationary states is characterised by a specific amount of energy called its energy level. An atom can have an amount of internal energy equal to any one of these possible energy levels, but it cannot have an energy intermediate between two levels.

The success of Bohr model is in explaining the spectral lines of atomic hydrogen and the Rydberg formula for the spectral emission lines. Bohr’s theory was the first to successfully account for the discrete energy levels of this radiation as measured in the laboratory. The emission line spectrum of hydrogen tells us that atoms of that element emit photons with only certain specific frequencies ƒ and hence certain specific energies equal to E = hƒ, where h = Planck’s constant = 6.63 x 10^-34 J.s and f = frequency of the photon.

Although Bohr’s atomic model is designed specifically to explain the hydrogen atom, his theories apply generally to the structure of all atoms. Subsequently, Bohr extended the model of hydrogen to give an approximate model for heavier atoms. Heavier atoms, like carbon or oxygen, have more protons in the nucleus, and more electrons to cancel the charge. Bohr’s idea was that each discrete orbit could only hold a certain number of electrons. In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons corresponded to stable “closed shells”. After that orbit is full, the next energy level would have to be used. This defines an electron shell, which is the set of allowed states and it gives the atom an electron shell structure.

Bohr’s Postulates

All these features of Bohr model of the atom can be summarized in Bohr’s Postulates:

1. Electrons in atoms orbit the nucleus.
2. An atom can exist only in certain specific energy states, in which an electron can reside without the emission of radiant energy.
3. Transmission between stationary states can occur only by jumping from one allowed orbit to another. This process produces/absorbs only a discrete quanta of electromagnetic radiation.
4. The angular momentum of a stationary electron is also quantised.

See also:

Atomic Theory

Pauli Exclusion Principle

The Pauli exclusion principle is the principle, formulated by Wolfgang Pauli in 1925, which states that:

No two electrons can occupy the same quantum-mechanical state within a given quantum system simultaneously.

In terms of the quantum numbers:

No two electrons in an atom can have the same values of all four quantum numbers.

It is impossible for two electrons of a multielectron atom to have the same values of the four quantum numbers:

• n – the principal quantum number,
• ℓ –  the angular momentum quantum number
• m_ℓ – the magnetic quantum number,
• m_s – the spin quantum number.

Mathematically this means the wavefunctions of the two particles must be antisymmetric which leads to the the probability amplitude of the wavefunction going to zero if the two fermionic particles are the same.

It is the Pauli exclusion principle that requires the electrons in an atom to occupy different energy levels instead of them all condensing in the ground state. The ordering of the electrons in the ground state of multielectron atoms, starts with the lowest energy state (ground state) and moves progressively from there up the energy scale until each of the atom’s electrons has been assigned a unique set of quantum numbers. This fact has key implications for the building up of the periodic table of elements.

It must be noted, chemical properties of atoms 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 as well as the Pauli exclusion principle. 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. As such, without the Pauli exclusion principle, there would be no chemistry.

This principle must be considered for any particles whose spin quantum number s is not zero or an integer. Therefore, this principle applies not only to electrons but also to protons and neutrons, all of which have half-integer spin.

Pauli Exclusion Principle and Nuclear Stability

Atomic nuclei consist of protons and neutrons, which attract each other through the nuclear force, while protons repel each other via the electromagnetic 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 (neutrons also obey the Pauli exclusion principle) 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.

The Pauli exclusion principle also influences the critical energy of fissile and fissionable nuclei. 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.

Pauli Exclusion Principle and Neutron Stars

A neutron star is the collapsed core of a large star (usually of a red giant). Neutron stars are the smallest and densest stars known to exist and they are rotating extremely rapidly. A neutron star is basically a giant atomic nucleus about 11 km in diameter made especially of neutrons. It is believed that under the immense pressures of a collapsing massive stars going supernova it is possible for the electrons and protons  to combine to form neutrons via electron capture, releasing a huge amount of neutrinos. Since they have some similar properties as atomic nuclei, neutron stars are sometimes described as giant nuclei. But be careful, neutron stars and atomic nuclei are held together by different forces.  A nucleus is held together by the strong force, while a neutron star is held together by gravitational force.

On the other hand, neutron stars are partially supported against further collapse by neutron degeneracy (via degeneracy pressure), a phenomenon described by the Pauli exclusion principle. In general, in a highly dense state of matter, where gravitational pressure is extreme, quantum mechanical effects are significant. Degenerate matter is usually modelled as an ideal Fermi gas, in which the Pauli exclusion principle prevents identical fermions from occupying the same quantum state. Similarly, white dwarfs are supported against collapse by electron degeneracy pressure, which is analogous to neutron degeneracy.

See also:

Atomic Theory

Atomic Nucleus

In physics, the atomic nucleus is the central part of an atom. In comparison to an atom, it is much more smaller and contains most of the mass of the atom. The atomic nucleus also contains all of its positive electric charge (in protons), while all of its negative charge is distributed in the electron cloud.

The atomic nucleus was discovered by Ernest Rutherford, who proposed a new model of the atom based on Geiger-Marsden experiments. These experiments were performed between 1908 and 1913 by Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford. These experiments were a landmark series of experiments by which scientists discovered that every atom contains a nucleus (whose diameter is of the order 10^-14m) where all of its positive charge and most of its mass are concentrated in a small region called an atomic nucleus. In Rutherford’s atom, the diameter of its sphere (about 10^-10 m) of influence is determined by its electrons. In other words, the nucleus occupies only about 10^-12 of the total volume of the atom or less (the nuclear atom is largely empty space), but it contains all the positive charge and at least 99.95% of the total mass of the atom.

After discovery of the neutron in 1932 by the English physicist James Chadwick, models for a nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko and Werner Heisenberg.

All matter except dark matter is made of molecules, which are themselves made of atoms. The atoms consist of two parts. An atomic nucleus and an electron cloud, which are bound together by electrostatic force. The nucleus itself is generally made of protons and neutrons but even these are composite objects. Inside the protons and neutrons, we find the quarks.

Inside the atomic nucleus, the residual strong force, also known as the nuclear force, acts to hold neutrons and protons together in nuclei. In nuclei, this force acts against the enormous repulsive electromagnetic force of the protons. The term residual is associated with the fact, it is the residuum of the fundamental strong interaction between the quarks that make up the protons and neutrons. The residual strong force acts indirectly through the virtual π and ρ mesons, which transmit the force between nucleons that holds the nucleus together.

Properties of Nucleus

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.

Mass of Nucleus

As was written, almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. The mass of the nucleus is associated with the atomic mass number, which is the total number of protons and neutrons in the nucleus of an atom. 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 ¹²C.

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 thermodynamics/thermodynamic-properties/what-is-mass-and-weight/standard-kilogram/”>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 ¹²C 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, ⁶³Cu (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:

1. 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 ¹²C with equal numbers of protons and neutrons.
2. 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 = mc². For ⁶³Cu the atomic mass is less than 63 so this must be the dominant factor.

Note that, it was found the rest mass of an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons and electrons. Mass was no longer considered unchangeable in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (E=mc²), this binding energy is proportional to this mass difference and it is known as the mass defect.

Radius and Density of Atomic Nucleus

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

r = r₀ . A^1/3

where r₀ = 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 πr² or 4.5×10⁻³⁰ m² for hydrogen nuclei or 1.74×10⁻²⁸ m² for ²³⁸U nuclei.

Nuclear density is the density of the nucleus of an atom. It is the ratio of mass per unit volume inside the nucleus. Since atomic nucleus carries most of atom’s mass and atomic nucleus is very small in comparison to entire atom, the nuclear density is very high.

The nuclear density for a typical nucleus can be approximately calculated from the size of the nucleus and from its mass. For example, natural uranium consists primarily of isotope ²³⁸U (99.28%), therefore the atomic mass of uranium element is close to the atomic mass of ²³⁸U isotope (238.03u). Its radius of this nucleus will be:

r = r₀ . A^1/3 = 7.44 fm.

Assuming it is spherical, its volume will be:

V = 4πr³/3 = 1.73 x 10^-42 m³.

The usual definition of nuclear density gives for its density:

ρ_nucleus = m / V = 238 x 1.66 x 10^-27 / (1.73 x 10^-42) = 2.3 x 10¹⁷kg/m³.

Thus, the density of nuclear material is more than 2.10¹⁴ times greater than that of water. It is an immense density. The descriptive term nuclear density is also applied to situations where similarly high densities occur, such as within neutron stars. Such immense densities are also found in neutron stars.

Excited Nucleus – Nuclear Resonance

Highly excited nuclei formed by the combination of the incident particle and target nucleus are known as nuclear resonances. If a target nucleus X is bombarded with particles a, it is sometimes observed that the ensuing nuclear reaction takes place with appreciable probability only if the energy of the particle a is in the neighborhood of certain definite energy values. These energy values are referred to as resonance energies. The compound nuclei of these certain energies are referred to as nuclear resonances. Resonances are usually found only at relatively low energies of the projectile. The widths of the resonances increase in general with increasing energies. At higher energies the widths may reach the order of the distances between resonances and then no resonances can be observed.

See also:

Atom

Mass of Nucleus – Atomic Mass Unit

As was written, almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. The mass of the nucleus is associated with the atomic mass number, which is the total number of protons and neutrons in the nucleus of an atom. 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 ¹²C.

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 thermodynamics/thermodynamic-properties/what-is-mass-and-weight/standard-kilogram/”>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 ¹²C 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, ⁶³Cu (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:

1. 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 ¹²C with equal numbers of protons and neutrons.
2. 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 = mc². For ⁶³Cu the atomic mass is less than 63 so this must be the dominant factor.

Note that, it was found the rest mass of an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons and electrons. Mass was no longer considered unchangeable in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (E=mc²), this binding energy is proportional to this mass difference and it is known as the mass defect.

See also:

Atom

Radius and Density of Atomic Nucleus

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

r = r₀ . A^1/3

where r₀ = 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 πr² or 4.5×10⁻³⁰ m² for hydrogen nuclei or 1.74×10⁻²⁸ m² for ²³⁸U nuclei.

Nuclear density is the density of the nucleus of an atom. It is the ratio of mass per unit volume inside the nucleus. Since atomic nucleus carries most of atom’s mass and atomic nucleus is very small in comparison to entire atom, the nuclear density is very high.

The nuclear density for a typical nucleus can be approximately calculated from the size of the nucleus and from its mass. For example, natural uranium consists primarily of isotope ²³⁸U (99.28%), therefore the atomic mass of uranium element is close to the atomic mass of ²³⁸U isotope (238.03u). Its radius of this nucleus will be:

r = r₀ . A^1/3 = 7.44 fm.

Assuming it is spherical, its volume will be:

V = 4πr³/3 = 1.73 x 10^-42 m³.

The usual definition of nuclear density gives for its density:

ρ_nucleus = m / V = 238 x 1.66 x 10^-27 / (1.73 x 10^-42) = 2.3 x 10¹⁷kg/m³.

Thus, the density of nuclear material is more than 2.10¹⁴ times greater than that of water. It is an immense density. The descriptive term nuclear density is also applied to situations where similarly high densities occur, such as within neutron stars. Such immense densities are also found in neutron stars.

See also:

Atom

Nuclear Shell Model

In nuclear physics, the nuclear shell model is a theoretical model to describe the atomic nucleus. The nuclear shell model was proposed by Dmitry Ivanenko in 1932 and further developed independently by several physicists such as Maria Goeppert-Mayer, Eugene Paul Wigner and J. Hans D. Jensen in 1949. It must be noted this model is based on the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels.

The nuclear shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a filled shell results in greater stability. Nucleons are added to shells which increase with energy that orbit around a central potential. In the atomic shell model the central potential around which the electrons orbit is generated by the nucleus. Nucleons are added to shells which increase with energy that orbit around a central potential.

In comparison to atomic shell model, the atomic nucleus governed by two different forces. The residual strong force, also known as the nuclear force, acts to hold neutrons and protons together in nuclei. In nuclei, this force acts against the enormous repulsive electromagnetic force of the protons. The term residual is associated with the fact, it is the residuum of the fundamental strong interaction between the quarks that make up the protons and neutrons. The strong interaction is very complicated interaction, because it significantly varies with distance. At distances comparable to the diameter of a proton, the strong force is approximately 100 times as strong as electromagnetic force. At smaller distances, however, the strong force between quarks becomes weaker, and the quarks begin to behave like independent particles. In particle physics, this effect is known as asymptotic freedom.

With the enormous strong force acting between individual nucleons and with so many nucleons to collide with, how can nucleons orbit a central potential without interacting? This problem is explained by the Pauli exclusion principle, which states that two fermions cannot occupy the same quantum state. In other words, the interaction will not occur, if the higher energy shells are fully occupied and the energy imparted to the nucleon during the collision is insufficient to promote the nucleon to an unfilled orbit. As a result, the nucleons orbit independent of one another.

The nuclear shell model was able to describe many phenomena like the magic numbers, the ground state spin and parity etc..

Magic Numbers of Protons and Neutrons

A magic number is a number of nucleons in a nucleus, which corresponds to complete shells within the atomic nucleus. Atomic nuclei consisting of such a magic number of nucleons have a higher average binding energy per nucleon than one would expect based upon predictions such as the mass formula of von Weizsaecker (also called the semi-empirical mass formula – SEMF) and are hence more stable against nuclear decay. Magic numbers are predicted by the nuclear shell model and are proved by observations that have shown that there are sudden discontinuities in the proton and neutron separation energies at specific values of Z and N. These correspond to the closing of shells (or sub-shells). Nuclei with closed shells are more tightly bound than the next higher number. The closing of shells occurs at Z or N = 2, 8, 20, 28, (40), 50, 82, 126. It is found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. Nuclei which have both neutron number and proton number equal to one of the magic numbers can be called “doubly magic“, and are found to be particularly stable.There are further special propertis of nuclei, which have a magic number of nucleons:

1. Higher abundance in nature. For example, helium-4 is among the most abundant (and stable) nuclei in the universe.
2. The stable elements at the end of the decay series all have a “magic number” of neutrons or protons. The nuclei He-4, O-16, and Pb-208 (82 protons and 126 neutrons) that contain magic numbers of both neutrons and protons are particularly stable. The relative stability of these nuclei is reminiscent of that of inert gas atoms (closed electron shells).
3. Nuclei with N = magic number have much lower neutron absorption cross-sections than surrounding isotopes.
4. These nuclei appear to be perfectly spherical in shape; they have zero quadrupole electric moments.
5. Magic number nuclei have higher first excitation energy.

Neutron Star as “Giant Nucleus”

A neutron star is the collapsed core of a large star (usually of a red giant). Neutron stars are the smallest and densest stars known to exist and they are rotating extremely rapidly. A neutron star is basically a giant atomic nucleus about 11 km in diameter made especially of neutrons. It is believed that under the immense pressures of a collapsing massive stars going supernova it is possible for the electrons and protons  to combine to form neutrons via electron capture, releasing a huge amount of neutrinos. Since they have some similar properties as atomic nuclei, neutron stars are sometimes described as giant nuclei. But be careful, neutron stars and atomic nuclei are held together by different forces.  A nucleus is held together by the strong force, while a neutron star is held together by gravitational force.

On the other hand, neutron stars are partially supported against further collapse by neutron degeneracy (via degeneracy pressure), a phenomenon described by the Pauli exclusion principle. In general, in a highly dense state of matter, where gravitational pressure is extreme, quantum mechanical effects are significant. Degenerate matter is usually modelled as an ideal Fermi gas, in which the Pauli exclusion principle prevents identical fermions from occupying the same quantum state. Similarly, white dwarfs are supported against collapse by electron degeneracy pressure, which is analogous to neutron degeneracy.

See also:

Atomic Nucleus