9.8 Quanta to Quarks

NOTE: The notes and worksheets for this topic are divided into two separate pages in order to keep download times within acceptable limits. To access the next page on this topic click on the link at the left marked 9.8 Continued.

INTRODUCTION:

In the early part of the twentieth century, many experimental and theoretical problems remained unresolved. Attempts to explain the behaviour of matter on the atomic level with the laws of classical physics were not successful. Phenomena, such as black-body radiation, the photoelectric effect, the emission of sharp spectral lines by atoms in a gas discharge tube, could not be understood within the framework of classical physics.

Between 1900 and 1930, a revolution took place and a new more generalized formulation called quantum mechanics was developed. This new approach was highly successful in explaining the behaviour of atoms, molecules and nuclei. As with relativity, quantum theory requires a modification of ideas about the physical world.

THE DEVELOPMENT OF ATOMIC MODELS

The first atomic theory of matter was introduced by the Greek philosopher Leucippus, born around 500 BC and his pupil Democritus, who lived from about 460 to 370 BC. However, the great success of the opposing “continuous matter” theory proposed by Aristotle (389-321 BC) ensured that the atomic model of matter took a back seat until about the 17^th Century AD. At this time the work and success of people like Copernicus, Galileo and Newton undermined the authority of Aristotle and allowed the atomistic view to be revived.

In the 19^th Century, John Dalton proposed an atomic model that allowed the first really quantitative study of the atom to be attempted. Later work by Sir J J Thomson and P Lenard led to further advances in our knowledge of the atom.

RUTHERFORD’S MODEL

In 1910 the New Zealand born physicist Ernest Rutherford, working in England, instructed two of his students, Hans Geiger and Ernest Marsden, to investigate alpha particle scattering from thin metal foils. What they discovered greatly enhanced our understanding of the atom.

Alpha particles are doubly charged helium nuclei and have a mass about 7500 times that of the electron and a velocity in this scattering experiment of about 1.6 x 10⁷ m/s. The existing model of the atom at that time (Thomson’s “Plum Pudding Model”) predicted that almost all of the alpha particles fired at a metal target would simply pass straight through the metal undeflected. To their great surprise, Geiger & Marsden found that a significant number of alpha particles were deflected by angles greater than 90^o. That is, the alpha particles were being reflected by the metal foil. Some even came back almost retracing their original path.

In 1911, Rutherford proposed his model of the atom, based on the results of many such scattering experiments. He proposed that the atom consisted mainly of empty space with a tiny, positively charged nucleus, containing most of the mass of the atom, surrounded by negative electrons in orbit around the nucleus like planets orbiting the sun. The electrons could not be stationary because if this were the case they would be attracted towards the positive nucleus and be neutralized. The Coulomb force of attraction between the positive nucleus and the negative electrons provided the necessary centripetal force to keep the electrons in orbit.

The Rutherford model was a great step forward in our understanding of atomic structure but it still had its limitations. Since the electrons were in circular motion, they would be experiencing centripetal acceleration and according to Maxwell’s Theory of Electromagnetism should be emitting electromagnetic radiation. This loss of energy would cause the electrons to gradually spiral closer and closer to the nucleus and to eventually crash into the nucleus. Thus, matter would be very unstable. This was clearly not the case. Also, Rutherford’s model could not explain the observed line spectra of elements. As electrons spiraled towards the nucleus with increasing speed, they should emit all frequencies of radiation not just one. Thus, the observed spectrum of the element should be a continuous spectrum not a line spectrum.

BOHR’S MODEL

Niels Bohr went to work with Rutherford in 1912. During the next two years he studied the Rutherford model of the atom. Bohr was inspired by the work of Max Planck on quantized energy and attempted to incorporate this idea into the atomic model to explain the discrepancies between the observed spectra of the elements and the spectra predicted on the basis of Rutherford’s atomic model.

As we saw in the “From Ideas To Implementation” topic, in 1900 Max Planck investigated the relationship between the intensity and frequency of the radiation emitted by very hot objects. Planck showed that the radiation from a hot body was emitted only in discrete quantities or “packets” called quanta. The energy, E, of each quantum was shown to be proportional to the frequency, f, of the radiation emitted:

where h = Planck’s constant = 6.63 X 10^-34 Js. This idea led directly to the belief that atoms could only absorb or emit energy in discrete quanta. Albert Einstein’s use of Planck’s quantisation idea to successfully explain the photoelectric effect added great support to this belief. So, Bohr was convinced that a successful atomic model had to incorporate this energy quantisation phenomenon.

Bohr’s thinking on a new atomic model was also guided by the work that had been done on the spectrum of hydrogen. Let us briefly examine firstly what is meant by the term spectrum and secondly the understanding of elemental spectra that existed at the time of Bohr’s work on the atom.

SPECTRA:

When an element such as hydrogen is heated to incandescence, or when it is ionized in a gas discharge tube, it emits visible light and other radiation that can be broken into its component parts using a spectroscope and a glass prism or a diffraction grating. The particular radiation emitted is known as the emission spectrum of that element and is unique to that element. When the emission spectrum of hydrogen is examined using a spectroscope, it is found to consist of four lines of visible light – a red line, a green line, a blue line and a violet line on a dark background. It can be shown that all elements produce emission line spectra rather than the continuous spectra predicted by the Rutherford model of the atom.

For a Practical Exercise on observing the visible lines in the hydrogen emission spectrum click on the following link - Hydrogen Spectrum Practical. Use your Browser's back arrow to get back here.

Another type of elemental spectrum is produced by passing white light through the cool gas of an element. The cool gas will absorb the same frequencies that it would otherwise emit if heated to incandescence. This spectrum is called the absorption spectrum of an element and consists of a continuous band of colours (different frequencies) with black lines present where particular frequencies have been absorbed by the cool gas. This spectrum is also unique to each element and is used to provide information on the elemental composition of stars.

The study of emission and absorption spectra of different elements provided much information towards the understanding of atomic structure. From 1884 to 1886 Johann Balmer, a Swiss school teacher, suggested a mathematical formula to fit the known wavelengths of the hydrogen emission spectrum:

where m is an integer with a different value for each line (m = 3, 4, 5, 6) & b is a constant with a value of 364.56 nm. This formula produces wavelength values for the hydrogen emission spectral lines in excellent agreement with measured values. This series of lines has become known as the Balmer series. Balmer predicted that there should be other series of hydrogen spectral lines and that their wavelengths could be found by substituting values higher than the 2 shown on the right hand side of the denominator in his formula.

In 1890, Johannes Rydberg produced a generalized form of Balmer’s formula for all wavelengths emitted from excited hydrogen gas:

where R = Rydberg’s constant = 1.097 x 10⁷ m^-1, n_f = an integer specific to a spectral series (eg for the Balmer series n_f = 2) and n_i = 2, 3, 4, ……

Gradually, other series of hydrogen emission lines besides the Balmer were found. The following table gives the details.

Name of Series	Date of Discovery	Region of EM Spectrum	Value of n_f	Value of n_i
Lyman	1906-1914	UV	1	2, 3, 4, …..
Balmer	1885	UV/Visible	2	3, 4, 5, …..
Paschen	1908	IR	3	4, 5, 6, …..
Brackett	1922	IR	4	5, 6, 7, …..
Pfund	1924	IR	5	6, 7, 8, …..

BOHR’S MODEL (Continued)

Although Rydberg’s equation was very accurate in its predictions of the wavelengths of hydrogen emission lines, for a long time no-one could explain why it worked – that is, the physical significance behind the equation. Bohr was the first to do so.

NOTE: The exact number and order of these postulates is not important. Some references will give two, some three & some four postulates for Bohr's model of the atom. What is important to know is the basic detail contained within the postulates. So, we could quickly summarise Bohr's postulates as: (1) Electrons orbit a central positive, nucleus in certain allowed, circular, orbits called stationary states from which they do not radiate energy. (2) Electrons only move from one state to another by absorbing or emitting exactly the right amount of energy in the form of a photon, whose energy is equal to the difference in energy between the initial & final states, DE = hf.

The first postulate retains the basic structure that successfully explains the results of the Rutherford alpha particle scattering experiments.

The second postulate was necessary to explain the observed atomic emission spectra of hydrogen. Only the separation of allowed orbits according to the second postulate gave the experimentally observed spectra. Clearly, Bohr’s study of the hydrogen spectrum was instrumental in the development of his model of the atom.

Clearly, the third postulate accounts for the observed stability of atoms. Bohr did not know why the stationary states existed; he simply assumed that they must because of the observed stability of matter.

The fourth postulate explains how atoms emit and absorb specific frequencies of electromagnetic radiation. An electron in its lowest energy state (called the ground state) can only jump to a higher energy state within the atom when it is given exactly the right amount of energy to do so by absorbing that energy from a photon of EM radiation of the right energy. Once the electron has jumped to the higher level, it will remain there only briefly. As it returns to its original lower energy level, it emits the energy that it originally absorbed in the form of a photon of EM radiation. The frequency of the energy emitted will have a particular value and will therefore be measured as a single emission line of particular frequency and therefore of particular colour if in the visible region of the EM spectrum.

Starting with these four postulates and using a mixture of classical and quantum physics, Bohr derived equations for: (i) the velocity of an electron in a particular stationary state; (ii) the energy of an electron in a particular stationary state; (iii) the energy difference between any two stationary states; (iv) the ionisation energy of hydrogen; (v) the radii of the various stationary states; (vi) the Rydberg constant; and (vii) the Rydberg equation for the wavelengths of hydrogen emission spectral lines.

In successfully deriving the Rydberg equation from his basic postulates, Bohr had developed a mathematical model of the atom that successfully explained the observed emission spectrum of hydrogen and provided a physical basis for the accuracy of the Rydberg equation. The physical meaning of the Rydberg equation was at last revealed. The n_f and n_i in the equation represented the final and initial stationary states respectively of the electron within the atom. The hydrogen emission spectrum consists only of particular wavelengths of radiation because the stationary states or energy levels within every hydrogen atom are separated by particular set distances, as described by the second postulate. The value of the Rydberg constant calculated by Bohr was in excellent agreement with the experimentally measured value.

Bohr’s atomic model led to a couple of useful ways of representing the quantum jumps of electrons involved in each of the different series of the hydrogen emission spectrum. These are shown below.

The following schematic diagram shows the possible transitions of an electron in the Bohr model of the hydrogen atom (first 6 orbits only).

The diagram below is an energy level diagram for the hydrogen atom. Possible transitions between energy states are shown for the first six levels. The dashed line for each series indicates the series limit, which is a transition from the state where the electron is completely free from the nucleus (n = infinity). The energies shown are the ionisation energies for electrons in each energy level. This is the energy that must be supplied to remove the electron in a given energy level from the atom. These energies are thus written as negative values.

Note: As you will see when you answer question 7 on the Bohr Worksheet, the energy of an electron in the n-th Bohr orbit is proportional to 1/n². Even though it is beyond the scope of the syllabus, it is worth stating that this implies that the gaps between successive higher energy levels get smaller and smaller (in terms of energy values) as indicated in the above diagram. In terms of spatial arrangement of Bohr orbits, however, the radius of the n-th Bohr orbit is proportional to n². So, spatially, the distance between successive higher orbits gets larger and larger. Taken together, these two facts make good sense, since the further the electron is from the nucleus, the less tightly it is held and therefore the less energy is required to move the electron from one energy level to a higher one.

LIMITATIONS OF THE BOHR MODEL:

In reality Bohr’s model was a huge breakthrough in our understanding of the atom. For his great contribution to atomic theory Bohr was awarded the 1922 Nobel Prize in Physics. As with any scientific model, however, there were limitations. The problems with the Bohr model can be summarised as follows:

WORKSHEET ON BOHR’S MODEL

1. State the four postulates used by Bohr to explain the nature of the atom.

2. Draw a sketch of the Bohr model of the atom, clearly labelling the electronic transitions responsible for the four visible lines in the hydrogen emission spectrum (H_a, H_b, H_g, & H_d ).

3. The red line within the Balmer series has a wavelength of 6560Å. Identify the initial and final stationary states corresponding to this transition within the hydrogen atom?

4. Calculate the wavelengths of each of the visible lines in the Balmer series for hydrogen.

5. Determine the frequency of the radiation emitted when an electron in a hydrogen atom undergoes a transition from the n_i = 2 energy level to the n_f = 1 level. Calculate the energy emitted by the electron in making this transition.

6. An electron in a hydrogen atom drops from stationary state n = 2 of binding energy 5.43 x 10^-19J to stationary state n = 1 of binding energy 21.56 x 10^-19J. Determine the energy emitted by the electron in making this transition.

7. Starting with the Rydberg equation, derive equations for the frequency and energy of the radiation emitted when an electron in a hydrogen atom undergoes a transition from stationary state n_i to stationary state n_f. Hence write an equation for the energy of an electron in the n-th stationary state of the hydrogen atom.

THE de BROGLIE MODEL

In 1924, Louis de Broglie, a French physicist, suggested that the wave-particle dualism that applies to EM radiation also applies to particles of matter. He proposed that every kind of particle has both wave and particle properties. Hence, electrons can be thought of as either particles or waves.

De Broglie reasoned that just as photons of EM energy have a momentum associated with their wavelength (p = h / l), particles of matter should have a wavelength associated with their momentum:

where p = momentum of particle, m = mass of particle, v = velocity of particle and h = Planck’s constant.

The impact of de Broglie’s proposal was far reaching. Its immediate impact was to provide a physical interpretation of the Bohr quantisation of stationary states within an atom. Its ongoing impact was to provide a new way of describing the nature of matter, which assisted greatly in the development of quantum mechanics. Erwin Schrodinger in 1926 used de Broglie’s ideas on matter waves as the basis of his wave mechanics, one of several equivalent formulations of quantum mechanics.

Let us examine how de Broglie’s matter wave proposal explains the Bohr quantisation of stationary states. Bohr’s second postulate states that:

An electron can occupy only certain allowed orbits or stationary states for which the orbital angular momentum, L, of the electron is an integral multiple of Planck’s constant divided by 2p. Mathematically, this can be written as:

De Broglie proposed that Bohr’s allowed orbits corresponded to radii where electrons formed standing waves around the nucleus. The condition for a standing wave to form would be that a whole number, n, of de Broglie wavelengths must fit around the circumference of an orbit of radius r.

Since (mvr) is the correct expression for the orbital angular momentum, L, of the electron in orbit around the nucleus, de Broglie had succeeded in showing that Bohr’s allowed orbits (or stationary states) are those for which the circumference of the orbit can contain exactly an integral number of de Broglie wavelengths. Thus, as shown in the figure below, the first stationary energy state (n = 1) corresponds to an allowed orbit containing one complete electron wavelength; the second stationary state corresponds to an allowed orbit containing two complete electron wavelengths; and so on.

De Broglie was then able to explain the stability of electron orbits in the Bohr atom. When an electron is in one of the allowed orbits or stationary states, it behaves as if it is a standing wave, not a particle experiencing centripetal acceleration. Thus, the electron does not emit EM radiation when it is in a stationary state within the atom.

Experimental confirmation of de Broglie’s proposal on matter waves was achieved in 1927 by Clinton Davisson and Lester Germer in the USA and by George Thomson in Scotland. Davisson and Germer conducted an experiment in which electrons in an electron beam produced the same diffraction pattern as X-rays when they were scattered by a small crystal of nickel.

As you will recall, diffraction is the name given to the phenomenon in which a wave spreads out as it passes through a small aperture or around an obstacle. Diffraction patterns are formed when the diffracted waves interfere with one another to produce light and dark bands on a screen or piece of film. Diffraction patterns are most intense when the size of the aperture or obstacle is comparable to the size of the wavelength of the wave. The electrons in the Davisson & Germer experiment were scattered in specific directions, which could only be explained by treating the electrons as waves with a wavelength related to their momentum by the de Broglie relation. Particles would have bounced off the nickel in all directions randomly.

The following is a diagram of the apparatus used by Davisson & Germer. Electrons from filament F are accelerated by a variable potential difference V. After scattering from the nickel crystal, they are collected by the detector D. D can be moved to measure the scatter yield at any angle.

Ever wondered what the basic difference is between classical physics and quantum mechanics? Well, take a detour for a minute and investigate the paradox of Schrodinger's Cat. This link takes you to the Brainteasers page of this site & to the description of the Schrodinger's Cat paradox that resides there. After reading the paradox, follow the links to the discussion of the paradox and then, if you are still interested, to a description of the domain of quantum mechanics itself. Use the back button of your browser to get back here. Go to Schrodinger's Cat.

WORKSHEET ON DE BROGLIE

1. Determine the de Broglie wavelength of the matter wave associated with a cricket ball of mass 0.175 kg and velocity 23.6 m/s. Use the answer to this question to explain why we do not observe the matter waves associated with macroscopic objects. (l = 1.6 x 10^-34 m)

2. Calculate the de Broglie wavelength of an electron travelling at 10⁶ m/s. (Mass of electron is 9.11 x 10^-31 kg.) Use the answer to this question to explain why we would expect to observe the effects of the matter waves associated with electrons. Give one example of these effects. (l = 7.3 x 10^-10 m)

3. A proton is travelling at a speed of 1.5 x 10⁷ m/s. Determine the de Broglie wavelength of the proton, given its mass is 1.67 x 10^-27 kg. (2.6 x 10^-14 m)

4. Calculate the momentum of a neutron if it has a de Broglie wavelength of 1.59 x 10^-13 m. (4.17 x 10^-21 Ns)

5. Determine the speed of the neutron in Q.4, given that the mass of a neutron is 1.67 x 10^-27 kg. (2.50 x 10⁶ m/s)

6. An electron volt (eV) is an energy unit equivalent to the work done when an electron is moved through a potential difference of 1 volt. If an electron has a kinetic energy of 100 eV, what is its associated de Broglie wavelength? (Charge on an electron is 1.6 x 10^-19 C) (1.2 x 10^-10 m)

7. Assess the contributions made by Heisenberg and Pauli to the development of atomic theory.

Once you have done that, check out the "Beam Me Up Scotty" section on the Brainteasers page. It's an interesting exercise on Heisenberg's Uncertainty Principle, quantum entanglement and quantum teleportation.

THE ELECTRON MICROSCOPE - Not Examinable

Note that this section on the Electron Microscope is no longer required by the Syllabus. As it is quite interesting, it has been left here as extension material.

The smallest object we can see with the unaided eye is about 0.1 mm. With a good light microscope we can magnify the image of the object up to about 1500 times but we are still limited in the detail we can see by the resolving power of the microscope. The resolving power of a microscope is a measure of its ability to distinguish clearly between two points very close together. The resolving power, a, is measured by the angular separation of two point sources that are just detectably separated by the instrument. The smaller this angle, the greater the resolving power.

where l is the wavelength of the light used and D is the diameter of the objective lens in metres. a is measured in radians (1 radian = 57.3^o approx.). The resolving power can easily be translated into a distance in metres from the geometry of the situation.

The best resolving power we can achieve with a light microscope is around 0.2 mm. Points closer together than this cannot be distinguished clearly as separate points. Clearly, if we could reduce the wavelength of the radiation used to view the sample, we could increase the resolution obtainable. De Broglie’s work on matter waves provided a theoretical means of achieving this wavelength reduction. De Broglie suggested that the matter waves associated with electrons accelerated through large potential differences would have wavelengths of the order of 0.5 nm. This is 1000 times smaller than the wavelength of green light (middle of the visible spectrum).

Max Knoll and Ernst Ruska constructed the first electron microscope around 1930. It used the wave characteristics of a beam of electrons rather than light to study objects too small for conventional light microscopes.

2. An image producing system consisting of magnetic lenses and metal apertures to confine and focus the electron beam, pass it through, or over the surface of, the specimen and create a magnified image.

3. An image viewing & recording system – usually photographic plates or a fluorescent screen.

4. A vacuum pump to keep the microscope under high vacuum – air molecules easily deflect electrons from their paths.

Before looking more closely at the different types of electron microscopes, it is worth stressing that the development of the electron microscope has had a massive impact on knowledge and understanding in many fields of science. Modern electron microscopes can view detail at the atomic level, 0.1 nm resolution (1000 times better than the light microscope) at up to a million times magnification.

For excellent diagrams & information on both TEM’s & SEM’s see the links below, as well as those on my Useful Links page in the From Quanta To Quarks section.

The Transmission Electron Microscope - Not Examinable

In the transmission electron microscope (TEM), electrons are transmitted through a thinly sliced specimen and form an image on a fluorescent screen or photographic plate. Those areas of the sample that are more dense will transmit fewer electrons (ie will scatter more electrons) and will therefore appear darker in the image. TEM’s can magnify up to one million times and are used extensively in Biology and Medicine to study the structure of viruses and the cells of animals and plants. The following diagram shows the basic structure of a TEM.

The Scanning Electron Microscope - Not Examinable

In the scanning electron microscope (SEM), the beam is focussed to a point and scanned over the surface of the specimen. Detectors collect the backscattered and secondary electrons coming from the surface and convert them into a signal that in turn is used to produce a realistic, three-dimensional image of the specimen. The detector receives back fewer electrons from depressions in the surface and therefore lower areas of the surface appear darker in the image.

SEM’s require the specimen to be electrically conducting, so specimens are coated with a thin layer of metal (often gold) prior to scanning. SEM’s can magnify up to around one hundred thousand times or more and are used extensively in many areas (Biology, Medicine, Physics, Chemistry, Engineering) to study the 3-D structure of surfaces from metals and ceramics to blood cells and insect bodies. A special type of SEM called the Scanning Tunnelling Electron Microscope (STM) uses a quantum mechanical effect called tunnelling to provide a 3-D image of the network of atoms or molecules on the surface of a specimen.

Magnetic Lenses In Electron Microscopes - Not Examinable

Many different types of lenses can be used to focus and control the electron beam in electron microscopes. Electrostatic lenses employ electric fields only to accomplish this task. Most lenses used, however, are magnetic in nature. Magnetic lenses can be either electromagnetic or permanent magnets. In either case, these lenses refract (bend) electron beams in a manner analogous to a biconvex glass lens refracting beams of light (photons).

An electromagnetic lens usually consists of a solenoid (coil), surrounded by a soft iron casing (shroud) to enhance the field and can also include soft iron pole pieces that further concentrate the field. Electrons travelling at angles to the field lines experience magnetic forces, which cause them to spiral around the field lines as they move through the magnetic lens. The radius of the spiral depends on the field strength. Electrons that enter the lens along the central axis of the solenoid will initially diverge due to repulsion from other electrons and then be refocused further along the lens. The current supplied to the solenoid controls the effective “shape” of the lens and therefore the focal length of the lens. The focal point, called the cross-over point, is where the electrons converge to a sharp focus. The diagram below shows an electromagnetic lens.

Both electromagnetic and permanent magnetic lenses can be designed to produce various different shaped magnetic fields as required. The diagram below shows a triangular magnetic field of uniform intensity. The triangular shape means that electrons that are furthest from the axis are in the magnetic field for more time and are consequently refracted at a greater angle than electrons near the axis. This results in the electrons in the beam being focused to a point on the axis beyond the magnetic lens.

If a rectangular shaped field is used, the electron beam can be focused by making the field non-uniform as shown below. Since the magnetic field strength increases with distance from the axis, the electrons near the axis are refracted through small angles, while those further away from the axis experience greater angles of refraction.

MAJOR COMPONENTS OF THE NUCLEUS

The quest to discover the nature of the nucleus has occupied generations of physicists since Rutherford discovered its existence in 1911. Rutherford determined from his scattering experiments that the nucleus was of the order of 10^-14 m in diameter. This turned out to be about 1/10 000 of the diameter of the atom, determined by Max von Laue in 1912 using X-ray diffraction to be 10^-10 m. Rutherford showed that the nucleus contained all of the positive charge of the atom and most of the atom’s mass. Henry Moseley, a graduate student working with Rutherford, found a direct correlation between an element’s position in the Periodic Table and its nuclear charge and also discovered that the total charge on a nucleus was equal to the total charge of the orbiting electrons in a neutral atom. By 1914 scientists accepted that a hydrogen ion (a hydrogen atom which has lost its electron) consisted of a singly charged particle. Rutherford named this the proton.

The Proton-Electron Model

A possible structure for the nucleus was then suggested by Rutherford. It was known in 1914 that an atom such as fluorine (atomic number 9) for example, had a mass equivalent to 19 protons but a charge of only 9 protons. So Rutherford suggested that the nucleus contained protons and electrons to balance the charge discrepancy. A fluorine nucleus would therefore contain 19 protons and 10 electrons – a total charge of 9 protons and total mass of 19 protons (electron mass being negligible compared to the proton mass).

This model was called the Proton-Electron Model. In general, a nucleus contained A protons and (A – Z) electrons, where A is called the mass number and Z the atomic number of the nucleus. This model could explain how a-particles & b-particles (electrons) could be emitted from some radioactive nuclei but problems arose:

u Quantum number anomalies arose with the spin of electrons and protons within the nucleus.

u Heisenberg’s Uncertainty Principle suggested that electrons could not be confined within the nucleus.

The Neutron

In a lecture in 1920, Rutherford suggested that a proton and an electron within the nucleus might combine together to produce a neutral particle. He named this particle the neutron. Experimental difficulties associated with the detection of a neutral particle greatly hindered the research. In 12 years of searching, no such particle was found.

In 1930, two German physicists, Bothe & Becker, bombarded the elements boron (B) and beryllium (Be) with a-particles. These elements, especially the Be, emitted a very penetrating form of radiation that was much more energetic than gamma-rays.

Frederic & Irene Joliot (Irene was the daughter of Marie Curie) found in 1932 that although this radiation could pass through thick sheets of lead, it was stopped by water or paraffin wax. They found that large numbers of very energetic protons were emitted from the paraffin when it absorbed the radiation. The Joliots assumed that the radiation must be an extremely energetic form of gamma radiation. In the same year, the English physicist, James Chadwick showed theoretically that gamma rays produced by a-bombardment of Be would not have sufficient energy to knock protons out of paraffin, and that momentum could not be conserved in such a collision between a gamma ray and a proton.

Chadwick repeated the Joliot’s experiments many times. He measured the energy of the radiation emitted by the Be and the energies (and therefore the velocities) of the protons coming from the paraffin. On the basis of its great penetrating power, Chadwick proposed that the radiation emitted from the Be was a new type of neutral particle – the neutron, as originally proposed by Rutherford. He then applied the conservation of energy and momentum laws to his experimental results and showed that the particles emitted from the Be had to be neutral particles with about the same mass as the proton. Chadwick had indeed discovered the neutron.

Chadwick explained that when the neutrons emitted from the Be collided with the light hydrogen nuclei in the paraffin, the neutron came to a sudden stop and the hydrogen nucleus (proton) moved off with the same momentum as the neutron had before the collision.

The Proton-Neutron Model

Following Chadwick’s discovery of the neutron, a new model of the nucleus was proposed. This model suggests that the nucleus consists of protons and neutrons. Together these particles are called the nucleons – particles that make up the nucleus. Protons and neutrons have approximately the same mass, which is about 1800 times that of the electron. Protons are positively charged and neutrons are neutral.

The number of protons in the nucleus is called the atomic number of the nucleus and corresponds to the position of the nucleus in the Periodic Table of Elements. The total number of protons and neutrons in the nucleus is called the mass number of the nucleus. Each nucleus can be represented by using nuclide notation. A nuclide is a nucleus written in the form:

where X = element symbol (eg Na, Co, U), Z = atomic number and A = mass number. Clearly, A = N + Z, where N = number of neutrons in nucleus. An alternate notation is to write the nucleus with its mass number after it – eg U-235 for uranium with a mass number of 235.

Isotopes of an element are atoms of that element varying in the number of neutrons present in their nuclei. Clearly, isotopes of the same element have the same atomic number but different mass numbers (and therefore slightly different masses). So, U-234, U-235 and U-238 are all isotopes of uranium – they all have 92 protons but differ in mass number.

The proton-neutron model of the nucleus is still the basic model used today. Many more nuclear particles have been found, however, and we will examine some of these a little later. For now, we turn our attention to the physical description of natural radioactivity.

Natural Radioactivity and Transmutation

Experimental work around the turn of the 20^th Century by Henri Becquerel (1896), Ernest Rutherford, Marie & Pierre Curie, Paul Villard and many other physicists led to the discovery of the three kinds of natural radiations – alpha particles, beta particles and gamma rays. These radiations were emitted naturally from certain elements (uranium, polonium, radium, actinium). Further, it was found that the emission of natural radiations by one element usually led to the production of a different element. For instance, radium was produced as a result of the radioactive decay of uranium.

This change of a parent nucleus into a different daughter nucleus is called nuclear transmutation. One element effectively changes into another element.

When transmutation occurs, the sum of the atomic numbers on the left hand side of the nuclear equation equals the sum of the atomic numbers on the right hand side. Likewise, the sum of the mass numbers on the left hand side of the nuclear equation equals the sum of the mass numbers on the right hand side.

Alpha Decay

where the helium-4 nucleus is the emitted alpha particle. Alpha decay occurs primarily among nuclei with atomic numbers greater than 83.

Beta Decay (The Weak Interaction)

Early attempts to explain beta decay assumed that an electron in the nucleus (Proton-Electron Model) was emitted in a process similar to that by which an alpha particle was emitted from a nucleus. One problem with this explanation, however, was that although all alpha particles emitted from a given species of nucleus had the same energy, beta particles emitted from a given species of nucleus seemed to have a continuous spectrum of energies.

James Chadwick, in some experiments conducted prior to World War I, used a Geiger Counter to study beta particles emitted from a source and then deflected by a uniform magnetic field. He found that the beta particles had a wide range of radii of curvature in the field, indicating that the beta particles had different velocities and therefore different energies. Similar experiments by many Physicists during the 1920’s and early 1930’s clearly indicated that during the beta decay of a particular nuclear species (eg Bi-210) electrons were emitted with a distribution of energies rather than with a distinct single value of energy.

The following graph shows the energy spectrum for electrons emitted during the decay of Bi-210. The intensity (vertical axis) shows the number of electrons emitted with each particular kinetic energy (horizontal axis).

Graphs such as this could not be explained. Why did the beta decay of a particular nuclear species produce many different beta particle emission energies? How was this possible when the decay process produced exactly the same daughter nucleus in each case?

Other experiments also suggested that the Law of Conservation of Energy was being violated. The total energy lost by the nucleus during b-decay was not equal to the total energy of the emitted b-particle.

The need to account for the energy distribution of electrons emitted in beta decay and to satisfy the Law of Conservation of Energy prompted Austrian physicist Wolfgang Pauli in 1930 to suggest that a neutral particle was emitted along with the b-particle. This particle would have no charge and no rest mass but would possess spin, energy and momentum.

Pauli believed that the emission of such a particle would successfully explain the spectrum of energies for emitted beta particles. For each beta emission, the total energy carried away from the decaying nucleus would be shared between the beta particle and the neutral particle emitted with it. So, when studying the beta decay of a sample, it would be expected that the beta particles emitted would have a range of energies depending on the energies of the neutral particles emitted with them. Clearly, Pauli’s idea also allows for the energy of reaction to be conserved, with both the beta particle and the neutral particle sharing the energy carried away from the decaying nucleus.

In 1934, Italian physicist Enrico Fermi named Pauli’s particle the neutrino (n), meaning “little neutral one” in Italian, and formulated a theory of b-decay using this particle. Fermi’s theory successfully explained all experimental observations. For instance, the shape of the energy curve shown above for Bi-210 can be predicted from the Fermi Theory of beta decay. Despite several ingenious attempts, the neutrino was not experimentally observed until 1956. In that year, two American Physicists, Cowan and Reines successfully identified the neutrino by detecting the products of a reaction that could only have been initiated by the neutrino. Basic details of this experiment are provided in “Nuclear Physics” by J Joyce & R Vogt (Brooks Waterloo, 1990).

b-decay is often referred to as the weak interaction because it is 10¹² times weaker than the strong nuclear force that holds the nucleus together.

b^--decay in which a neutron decays to produce a proton, an electron and an anti-neutrino

The electron and the anti-neutrino are emitted but the proton stays behind, thus increasing the atomic number by one.

The following equation for example, describes the spontaneous decay of C-14 in the upper atmosphere, as it is produced by bombardment of nitrogen by neutrons in cosmic rays:

b⁺-decay in which a positron (positive electron) is emitted after a proton decays to produce a neutron, a positron and a neutrino.

The following equation for example, describes the b-decay of the artificially produced N-13 nucleus:

(As an aside, it is worth noting that the distinguishing feature between the neutrino & anti-neutrino is their helicity. The anti-neutrino has its spin angular momentum parallel to its linear momentum – it has a right hand screw helicity. The neutrino has its spin angular momentum anti-parallel to its linear momentum – it has a left hand screw helicity.)

Gamma Emission

This usually accompanies a or b-decay. A nucleus de-excites by emitting a high-energy gamma ray photon. This is not a transmutation.

THE STRONG NUCLEAR FORCE

It is known that there is an electrostatic Coulomb repulsion force between any two like charges. So, in the case of protons in the nucleus, there must be some sort of force that holds the protons together. At first we might be tempted to suggest that the gravitational attraction that exists between all bodies possessing mass is responsible for holding the protons together. However, if we evaluate the relative contributions of the electrostatic and gravitational forces between protons, we find that the gravitational force is millions of times smaller than the electrostatic force. Thus, there must be another force at work.

The force responsible for holding all nucleons together is the strong nuclear force. The graph below shows the strong nuclear force between nucleons as a function of the separation of the nucleons.

u At typical nucleon separation (1.3 x 10^-15m) it is a very strong attractive force (10⁴ N).

u At much smaller separations between nucleons the force is very powerfully repulsive.

u Beyond about 1.3 x 10^-15m separation, the force quickly dies off to zero.

u The much smaller Coulomb force between protons has a much larger range and becomes the only significant force between protons when their separation exceeds about 2.5 x 10^-15m.

u The strong nuclear force is not connected with charge. Proton-proton, proton-neutron and neutron-neutron forces are the same. (The force between protons, however, must always be modified by the Coulomb repulsion between them.)

COMMON UNITS USED IN NUCLEAR PHYSICS

For convenience, nuclear physicists usually use the atomic mass unit (u) as the unit of mass and the electron volt (eV) or Mega electron volt (MeV) as the unit of energy. These are defined as follows.

ATOMIC MASS & ATOMIC MASS UNIT

The present standard atom is the atom of the commonest isotope of carbon, C-12. By definition this isotope of carbon has a mass of 12.0000 atomic mass units (u) exactly. Thus, since the mass of one C-12 atom is 1.9924 x 10^-26 kg (by mass spectrograph measurements), we have:

THE ELECTRON VOLT

The electron volt is the amount of energy gained by an electron as it is accelerated through a potential difference of one volt.

Clearly, using Einstein’s equation for the equivalence of mass and energy we have:

MASS DEFECT AND BINDING ENERGY

The experimentally measured mass of any nucleus is less than the sum of the masses of its constituent protons and neutrons.

For example, let us consider an atom of the commonest isotope of chlorine, Cl-35.

The actual mass of this atom, determined by experiment, is 34.980175 u.

The combined mass of the constituent particles may be determined as follows:

The difference in mass is called the mass defect of the atom (or nucleus, if we are dealing with the nucleus only). In this case, the mass defect is about 0.309 u or 5.13 x 10^-28 kg.

This small mass has been converted into the binding energy of the nucleus (the energy holding the nucleus together). The mass defect of a nucleus can therefore be defined as the mass equivalent of the binding energy of the nucleus. The amount of binding energy involved in this example is:

By definition, the binding energy of the nucleus is the energy needed to separate the nucleus into its constituent parts. When the nucleons come together to form the nucleus, they release the binding energy.

If we take the total binding energy of a nucleus and divide it by the total number of nucleons in the nucleus, we get a very good measure of how tightly each individual nucleon is held in the nucleus. This binding energy per nucleon figure is a very good measure of the stability of the particular nucleus. The higher the binding energy per nucleon, the more stable the nucleus. The diagram below shows the basic shape of the binding energy per nucleon versus mass number graph. This will be a useful tool for explaining nuclear fission and fusion a little later. (Note that the vertical axis has been drawn on the right for clarity.)

Note that the binding energy per nucleon is low for low mass number nuclei. This is because in such nuclei each nucleon is not uniformly surrounded and thus does not experience the full effects of the strong nuclear force. Most nuclei have binding energy per nucleon values between 7 and 9 MeV, with the highest value being that for Fe-56. For very high mass number nuclei the electrostatic repulsive forces between the protons result in a gradual decrease in binding energy per nucleon values.

NOTE: The rest of the notes and worksheets for this topic are found on the next page.