The nuclear model and particle experiments (Topic 8A)
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The term nucleon refers to particles inside the nucleus: protons and neutrons:
- The proton and the neutron have approximately the same mass.
- The proton has a charge of where is the elementary charge.
- The neutron has no charge.
An atom has no net charge and has the same number of protons as electrons.
For an element the nucleus of an atom of that element can be represented by nuclear notation.
Where:
- is the nucleon number: the total number of protons and neutrons. is also known as the mass number, since it gives the mass of the nucleus in units of the proton (or neutron) mass.
- is the proton number: the number of protons. is also known as the atomic number, since it determines the type of atom.
Isotopes are atoms of the same element that contain a different number of neutrons in their nuclei.
For example, carbon exists in three naturally occurring isotopes: carbon-12 (six neutrons), carbon-13 (seven neutrons) and carbon-14 (eight neutrons).
- Isotopes of a given element have the same number of protons and hence the same number of electrons, meaning they have the same chemical properties.
- Isotopes of a given element have different masses, meaning they have different physical properties, such as melting point and density.
When using nuclear notation, always label the element with the nucleon and proton numbers.
For example, carbon-14 has six protons and eight neutrons.
In 1911, Rutherford’s alpha-particle scattering experiment proved that atoms contain a small, positively charged nucleus at the centre.
An alpha particle is a helium-4 nucleus — consisting of two protons and two neutrons.
In Rutherford’s alpha particle scattering experiment, a beam of alpha particles was fired at a thin sheet of gold foil.
The angles at which the alpha particles were scattered by the gold foil were measured by a circular fluorescent screen.
The following observations were made from the alpha particle scattering experiment:
- Most of the alpha particles passed straight through the gold foil (approximately )
- A small fraction of the alpha particles were scattered by an angle greater than (approximately )
- An even smaller fraction of the alpha particles were deflected back toward the source (approximately )
The alpha-particle scattering experiment led Rutherford to the following conclusions about the atom:
- Most of the atom is empty space, since most of the alpha particles passed straight through.
- Most of the mass of the atom is concentrated in the nucleus since the alpha particles had a relatively high momentum and some were still reflected back toward the source.
- The nucleus was positively charged since it repelled the positive alpha particles that passed nearby and caused them to deflect.
The alpha particle scattering experiment determined the existence of the atomic nucleus:
- In 1917, Rutherford determined that the hydrogen nucleus was a single proton.
- Chadwick discovered the neutron in 1933.
- These discoveries led to the nuclear model of the atom, which features a nucleus at the centre surrounded by electrons.
- Niels Bohr later improved this simple model by suggesting that electrons orbit the nucleus at certain distances from the nucleus in electron shells.
The atomic nucleus consists of protons and neutrons:
- Protons are positively charged and neutrons have zero charge so the overall charge of the nucleus is always positive.
- Protons and neutrons have approximately the same mass.
In the nuclear model of the atom, it was determined that:
- Electrons orbit the nucleus in fixed energy levels.
- For every atom the number of electrons is always equal to the number of protons so the overall charge is zero.
- The mass of the electron is equal to approximately that of the proton.
In the alpha particle scattering experiment, Rutherford used the distribution of alpha particles to determine an upper limit on the radius of a gold nucleus as having magnitude
An alpha particle and a gold nucleus are both positively charged, so they repel each other.
When an alpha particle is fired directly at a gold nucleus, it will reach a point of closest approach and then travel back in the opposite direction.
When an alpha particle is fired directly at a positively-charged atomic nucleus, the distance of closest approach can be determined using the principle of conservation of energy.
The alpha particle has maximum kinetic energy when it is fired from a source.
As it travels closer toward the nucleus, the alpha particle experiences a stronger repulsion.
When the alpha particle reaches the point of closest approach, it comes to a stop so its kinetic energy is zero: all of its initial kinetic energy has been converted to electric potential energy
The decrease in kinetic energy is equal to its increase in electric potential energy, so that the sum of the two quantities remains constant and that energy is conserved.
Nuclear radii have now been measured as ranging between to
In the years following the alpha scattering experiment of 1911, X-ray diffraction measurements showed that the atomic radius is approximately five orders of magnitude larger than the nuclear radius.
If the nucleus of an atom were the diameter of a needle head, then the atomic radius would be the length of a football field.
Question walkthrough
Estimating Closest Approach in Alpha Scattering
Estimates the closest approach distance of an alpha particle fired at a gold nucleus by equating its kinetic energy to electric potential energy.
When a metal is heated, the free (delocalised) electrons within the metal gain kinetic energy. If enough thermal energy is supplied to a metal, the electrons at the surface gain enough kinetic energy to escape. This is known as thermionic emission.
The work function of a metal is the minimum kinetic energy an electron needs to escape the surface.
Thermionic emission occurs when the kinetic energy of the electrons on a metal’s surface exceeds the work function. For many metals, thermionic emission becomes significant at temperatures above
Electrons emitted via thermionic emission can be accelerated by an electric field. This is the principle of operation in an electron gun.
The diagram below shows the inner structure of an electron gun.
An electron gun produces a narrow beam of electrons:
- A coil is heated electrically, which in turn heats a metal cathode.
- Electrons are emitted from the cathode via thermionic emission.
- An electric field applied by a potential difference between the cathode and anode accelerates the electrons towards a cylindrical anode.
- Only the electrons directed at the hole in the anode can pass through, creating a narrow beam.
- The electrons move at constant velocity beyond the anode due to the absence of an electric field.
Electric fields can be used to accelerate charged particles. For example, linear accelerators utilise electric fields to accelerate protons to speeds approaching the speed of light.
The image below shows two oppositely charged plates with a proton placed between them:
The proton, being positively charged, moves downwards following the electric field lines. As the force is constant, the proton has constant acceleration.
When a charged particle travels into a uniform electric field at a right angle, the particle follows a parabolic trajectory.
The image below shows the motion of a positively charged particle entering a uniform electric field at a right angle:
It is important to note that the particle follows a curved trajectory whilst inside the field and continues to follow a straight path once it has left the field.
There is no horizontal acceleration since no horizontal force is being applied, thus the horizontal component of the velocity remains constant throughout the motion.
There is a vertical acceleration of the particle due to the electric field applying a vertically downward force on the particle. The particle initially has zero vertical velocity but upon exiting the field has acquired a vertical component of velocity .
A particle of charge travelling in a magnetic field experiences a magnetic force .
When the velocity vector of the particle is perpendicular to the magnetic field vector , the magnetic force acts on the particle perpendicularly to both and .
It is important to note that the particle will follow a circular trajectory because the magnetic force is perpendicular to the velocity. The magnetic force will always point toward the centre of the circle, acting as a centripetal force.
A variation of Fleming’s left-hand rule can be used on charged particles to determine the direction of the magnetic force acting on that particle.
For a positive charge, the index finger of the left hand points in the direction of the magnetic field, the middle finger in the direction of motion, and the thumb points in the direction of the magnetic force.
Follow the same process for a negative charge, but flip the direction of the magnetic force at the end.
It is important to recall and understand the proper use of Fleming’s left-hand rule and its variation.
The magnetic force, , is a centripetal force that acts on the charged particle. Hence, we can apply the formula for a centripetal force:
However, centripetal acceleration can be expressed in terms of speed, , and radius, , of circular motion as:
Finally, combining the previous two equations, we get:
Where:
- is the mass of the particle in ,
- its speed in and
- is the radius of the circle traversed by the particle, in .
Knowing that the magnetic force is also equal to:
The equation becomes:
Thus, the radius of the circle completed by the charged particle is:
The radius of the circle can be increased by:
- increasing the mass of the charge ,
- increasing the speed of the charge ,
- reducing the magnetic flux density , and
- reducing the quantity of charge .
Question walkthrough
Finding Radius of a Proton's Circular Path in a Field
Calculate the radius of the circular path of a proton moving at right angles to a uniform magnetic field, given its charge-to-mass ratio and speed.
Charge is always conserved in particle interactions and decays: total charge before = total charge after.
Charge comes in whole multiples of , so add up the charges on each side and check they match. If they don’t, the interaction is forbidden. This is the quickest first test on any proposed reaction.
An example of this is in decay, where a neutron (0) becomes a proton (+1), an electron (−1) and an antineutrino (0), giving , so charge balances.
In every particle interaction, total energy and total momentum are conserved:
- Energy includes kinetic energy and the rest energy of the particles (via ), so mass can convert into kinetic energy and back.
- Momentum is a vector. Conserve it separately in each direction. The products’ momenta must sum to the incoming momentum of the original particle(s).
Kinetic energy alone is not always conserved: some of it becomes the rest energy of new particles, which is why creating massive particles requires a minimum threshold energy.
Interpreting particle tracks relies on the motion of charged particles in a magnetic field. Only charged particles ionise the medium and leave a track; neutral particles (e.g. photons) leave a gap, inferred from where tracks appear or vanish.
- Radius: from , larger radius means greater momentum; as energy is lost shrinks, so tracks spiral inwards.
- Curvature direction gives the sign of charge. Opposite charges curve opposite ways in the same field.
- Apply conservation of charge, energy and momentum at each vertex. For example, in pair production a photon leaves no track, then creates two opposite tracks.
The wavelength of a particle is related to its momentum by the de Broglie equation, which states that:
where:
- is the momentum of the particle in
- is the Planck constant
- is the wavelength of the particle in
The de Broglie equation shows that the momentum of a particle is inversely proportional to its wavelength:
For particles travelling at the same speed, a greater mass results in a shorter wavelength.
The equation for the momentum of a particle in is:
Where:
- is the mass of the particle in and
- is the speed of the particle in
The de Broglie wavelength of a particle is inversely proportional to its momentum, and this can be observed in the electron diffraction experiment.
Slower electron acceleration by the electric field leads to lower momentum upon reaching the graphite grating. Consequently, the electrons exhibit a larger de Broglie wavelength, which is a closer match to the atomic spacing in the graphite. This results in increased diffraction and a broader interference pattern.
A greater momentum, which corresponds to faster electrons, leads to a smaller de Broglie wavelength. Consequently, the electrons diffract less, producing a narrower interference pattern.
Electrons moving near the speed of light have de Broglie wavelengths of which is small enough to determine the size of nuclear radii, with size of or by electron diffraction.
This correlates to the scales where the strong nuclear force, which binds the nucleons within an atom, is attractive.
The equation for the approximate radius of a nucleus has been determined by electron diffraction experiments.
Where:
- is the nucleon number of the nucleus,
- and is a constant.
Albert Einstein proposed that energy and mass are interchangeable and demonstrated this with his famous equation:
Where:
- is energy ,
- is mass , and
- is the speed of light in a vacuum .
The equation tells us that mass and energy are essentially the same thing and therefore equivalent, but have different forms. When we multiply mass by the speed of light squared, we get the amount of energy equivalent to that amount of mass.
An example of mass–energy equivalence is the annihilation of an electron–positron pair. When the two particles collide, they annihilate and produce two photons. The mass of the electron and positron is converted into two photons, i.e., energy.
When there is a change in the mass of an object or system, then there is an equivalent change in the amount of energy represented by Einstein’s equation:
A moving object possesses kinetic energy. Thus, the equation implies that the object’s mass when it is moving is greater than its mass while at rest (rest mass).
All particles have a corresponding antiparticle. They are similar in every aspect, except that they have opposite charges. When a particle and its corresponding antiparticle collide, they annihilate each other: all of the particle’s mass is converted into energy, typically in the form of photons.
An example of the process of annihilation is electron–positron annihilation:
The electron has an associated antiparticle known as the positron . The electron and the positron are similar in every aspect, except that they have the opposite charge. When an electron and positron collide, they annihilate and all their mass is transformed into energy in the form of two identical photons.
A process opposite to annihilation can occur, in which a photon transforms into a particle-antiparticle pair: this is known as pair production.
For example, a photon may create an electron–positron pair, provided the photon has an energy that is equal to at least twice the rest-mass energy of the electron i.e.
For pair production to occur, a photon must enter into the Coulomb field of a nearby atomic nucleus: it cannot happen in free space. This is because the reaction requires energy and momentum to be conserved.
In free space, the momentum of the electron and positron combined would not equal that of the photon’s initial momentum. Therefore, the nucleus absorbs some of the photon’s initial momentum, thus obeying conservation laws in the reaction.
Question walkthrough
Photon Energy from Electron-Positron Annihilation
Calculates the energy of each photon produced when an electron and positron annihilate, using E = mc² with the rest mass of an electron.
An electronvolt is defined as the kinetic energy gained by an electron after being accelerated through a potential difference of The kinetic energy gained by a charged particle accelerated through a potential difference is given by:
Where:
- is the magnitude of the charge of the particle in
- is the potential difference in
The electron charge is so one electronvolt is equal to:
Planck’s constant, is an extremely small quantity. The energy of a photon is equal to:
Therefore, the energies in of individual photons are extremely small. The electronvolt is a more convenient unit of energy for photons. An example of this is the energy in electronvolts of a red visible light photon of wavelength is , which is easier to read and use than .
It is important to note that the following units are commonly used as units of energy:
Mass and energy are linked by , so rearranging gives:
This means a mass can be expressed in units of energy divided by :
These units are convenient in particle physics as they avoid very small values in . To convert to SI units, write the energy in joules and divide by :
For example, an electron has a mass of about
Time passes at different rates for observers moving in relative motion. A clock moving relative to an observer appears to tick more slowly than a clock at rest with respect to that observer. This phenomenon is known as time dilation, a prediction of the theory of special relativity.
The proper time is the time interval measured by a clock (or observer) at rest relative to the events being observed. In this situation, a stationary observer measuring two events with a clock records the shortest possible time interval between those events.
It is important to note that since absolute motion doesn’t exist, a ‘stationary’ observer simply means someone at rest relative to the frame in which the events occur.
Imagine a high-speed train travelling at a speed close to that of light. A passenger on the train may measure their journey to last 2 minutes according to the train’s clock. This is the proper time. However, an observer standing on the platform may see the same journey take longer, because the train’s clock is moving relative to them.
An observer moving relative to the events taking place measures a longer time interval than the proper time, since the moving clock appears to run slower. The difference is usually negligible at everyday speeds but becomes significant as the relative speed approaches the speed of light.
Muons are unstable particles created in the Earth’s upper atmosphere that move toward the surface at speeds close to the speed of light. Their half-life at rest is very short so classically, almost all should decay before reaching the ground.
However, this is not the case; a considerably larger number of muons survive the journey than expected. These can be detected on the Earth’s surface, providing experimental evidence of time dilation.
The following experiment measures muon decay:
- Measure muon speed from rest in a non-inertial frame (expect ).
- Measure the muon count rate with a detector in the high atmosphere.
- Measure the muon count rate at ground level with a second detector.
- Compare count rates to calculate the muon survival number.
Experimental data indicate that far more muons survive than predicted by classical models based on their short half-life. Proper time is measured in the reference frame which is stationary relative to the events being measured. In this case, the event is the muons decaying, so proper time is measured in the muon’s frame of reference.
From the observer’s frame on Earth, time dilation appears to extend the muons’ lifetime, so more survive the journey. However, from the muon’s frame, it is at rest and measures the proper time for its decay. From the muon’s frame of reference, the journey actually takes less time than its half-life, which explains its survival.