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Atomic and Nuclear Physics

Photoelectricity

This is the current produced by the freeing ore escaping of electrons from a negatively charged

metal by exposing that metal to ultra-violet light. (Example: ultra-violet radiation ejects electron

from zinc while white light ejects electrons from sodium)

This process is called the photoelectric effect and the light (radiation) gives energy to the

electron in the surface atoms of the metal and enables them to break through the surface.

An experiment which demonstrated the nature of photoelectricity

The surface of the zinc plate is rubbed with emery paper until clean and bright. One of the zinc

plates is insulated and connected to the cap of a gold leaf electroscope and it is given a positive

charge by the induction (the process by which electric or magnetic state is obtained by exposure

to an electric or magnetic field)

Some of this charge spreads to the leaf which opens. In a dark room this positively charged zinc

plate is then exposed to ultra-violet light from a small lamp placed near it. The leaf stays open

which indicates that there is no loss of charge. The free electrons need much more energy to

leave the zinc plate because it is positively charged. The radiation cannot supply enough energy

and therefore no emission of electrons occurs, since the electrons are attracted back to the plate.

However, in the case of the negatively charged zinc plate: the leaf slowly falls because the

electroscope loses it charge. Since both the electrons and the zinc plate now carry a negative

charge, the electrons are repelled away from the zinc plate and are therefore easily emitted.

These electrons are called photoelectrons which indicates that the electron has been emitted

when light fell on the surface of the metal (N.B. These electrons are identical to any other

electrons)

The Phenomenon of Photoelectric Emission

The measurements and investigations have shown the following conclusions:

The kinetic energy (velocity) of the electrons emitted from an illuminated metal is

independent of the intensity of the light (i.e. The intensity of the radiation has no effect

on the kinetic energy of the emitted electrons)

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For a given metal, no photoelectrons are emitted if the light frequency (radiation

frequency) is below a certain value called the threshold frequency. This frequency is

defined as the minimum frequency of electromagnetic radiation for which photoelectric

emission occurs.

No electrons are emitted when the metal is illuminated by light (radiation) of a

wavelength longer than a specific wavelength called the threshold wavelength. Electrons

are emitted when the metal is being illuminated by light (radiation) which has a

wavelength lower than the threshold wavelength

The emission commences as soon as the surface starts to be irradiated.

The number of photoelectrons emitted per second from any given metal is proportional to

the intensity of the incident radiation. (The more intense the radiation the greater the

number of photoelectrons leaving the metal each second).

The emitted electrons have different kinetic energies (ranging from zero up to a

maximum value). Increasing the frequency of the incident radiation will increase the

energies of the electrons emitted and the also the maximum kinetic energy.

The Inability of Classical Physics (Wave Theory) to Explain Aspect of Photoelectric Effect

The photoelectric effect is due to electrons absorbing energy from the radiation and therefore

having the ability to overcome the attractive forces of the nuclei. The wave theory of light states

that the energy of the radiation is distributed evenly over the wavefront.

According to the wave theory, each electron on the surface of the metal will absorb an equal

share of the radiation. Hence the expectation is that the intensity of the radiation would be very

low and no electrons would gain the required amount of energy to escape the metal, or that a

significant amount of time would have lapsed before any electrons escape.

Neither of these two predictions are consistent with the observations seen. Furthermore,

increasing the intensity, increases the energy fall on the surface of the metal and an increase in

the energies of the emitted electrons would be expected. However this is also inconsistent with

the observations made.

In addition, the wave theory does not offer an explanation of the frequency being depend on the

kinetic energy of the electrons being emitted nor why there should be a minimum frequency at

which emission occurs.

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Planck’s Equation

Max Planck showed that the laws could be explained by assuming light and all other forms of

electromagnetic radiation are emitted in wave packets, with each wave packet being a short burst

of light energy from an atom (quantum theory).

He showed that when an atom emits light, its energy changes by specific allowed amounts.

When referring to the light energy, these packets of energy are called photons. (packets of

electromagnetic energy emitted by an atom).

The energy of a single photon is proportional to its frequency and therefore the following

equation can be stated:

𝐸 = ℎ𝑓

where E = photon energy

h = Planck’s constant (6.626 × 10-34 Js)

f = the frequency of the radiation

Note that :

𝑓 = 𝑐

𝜆

where c = speed of light (3× 108 m/s)

𝜆 = wave length

Work Function (W0)

The least or the minimum amount of work or energy necessary to take a free electron out of a

metal against the attractive forces of surrounding positive ions is called the work function of the

metal.

We can calculate the threshold frequency (f0) by using the following equation:

𝑓0 = 𝑊0

ℎ

And the corresponding maximum wavelength (𝜆0) is given by:

𝑊0 = ℎ𝑐

𝜆0

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The Electron-volt (eV)

This is the unit of energy. It is defined as the kinetic energy gained by an electron which is

accelerated by a potential difference of one volt.

1 𝑒𝑉 = 1.6 × 1019 𝐽

Einstein’s Photoelectric Equation

Einstein’s Theory comprises of the following statements

light of frequency (f) contains quanta of energy (hf)

light consists of particles called photons

the number of photons per unit area of cross section of a beam of light per second is

proportional to its intensity

the energy of a photon is proportional to its frequency but independent of its light

intensity

A simple equation can be obtained for the maximum kinetic energy of electrons liberated from

an illuminated metal, and we can state that:

The maximum kinetic energy of a photoelectron is equal to the energy gained by absorbing a

photon (hf) minus the work done (W0) to escape from the material.

𝐾. 𝐸.𝑚𝑎𝑥 = ℎ𝑓 − 𝑊0

1

2𝑚𝑣2 = ℎ𝑓 − 𝑊0

Therefore Einstein’s Photoelectric Equation is summed up by:

ℎ𝑓 = 𝐾. 𝐸.𝑚𝑎𝑥 + 𝑊0

OR

ℎ𝑓 = 1

2𝑚𝑣2 + 𝑊0

These equations show that the maximum kinetic energy depends only on the light frequency and

that K.E. max is unaffected by making the light brighter (light intensity)

𝐾. 𝐸.𝑚𝑎𝑥 = 1

2𝑚𝑒𝑣2

where 𝑚𝑒 = 9.1 × 10−31 𝑘𝑔 (mass of an electron)

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The Stopping Potential (Vs)

The stopping potential of a surface is the minimum positive potential which must be applied to a

surface to stop photoelectric emission at a particular light frequency.

𝑒𝑉𝑠 = ℎ𝑓 − 𝑊0

Where e = photon energy

W0 = Planck’s constant (6.626 × 10-34 Js)

Vs = the frequency of the radiation

If we make Vs the subject of the equation, we get that

𝑉𝑠 = ℎ𝑓

𝑒−

𝑊

𝑒

Since this equation is in the format of a straight line (y = mx + c), we can see that

y = Vs, x = f, c (y- intercept) = W/e (depends on the type of material used)

m (gradient) = h/e (not dependent on the type of material since h and e are fundamental

constants),

NOTE: 𝐾. 𝐸.𝑚𝑎𝑥 = 𝑒𝑉𝑠

Determining the Stopping Potential

X - photoelectric cell

C - photosensitive metal of large area

A - collecter of electrons in a vacuum

Y - potential divider arrangement for varying P.D. (V) between the anode, A and the cathode, C

d.c. ammeter - ammeter measuring the small current

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What happens?

As the incident light reaches C, electrons are released from the surface of C. Although these

electrons are negative in charge, as so is A, these electrons make their way to A because the

energy they possess upon being released allows them to surpass the repulsive force which they

experience from A.

This occurs for the initial value of V and since electrons are able to move some current in noted

at the d.c ammeter. As the p.d at A rises by the potential divider, A becomes more and more

negative and eventually the electrons are no longer able to surpass the repulsive force (their

energy is less than the force) and A is able to keep the electrons from flowing through out the

circuit. As a result of this no current flows through the circuit. As a result of this no current is

read at the d.c. ammeter. The value of V at which this phenomenon occurs is called the stopping

potential.

Wave-Particle Duality

The particle theory describes light as being made up of small particles moving linearly. This

theory is used to explain reflection and refraction. It basically states that light was made up of

tiny streams of particles, traveling at very high speeds, in straight line. However, could not be

used to explain diffraction of light.

Therefore another model, the wave theory was developed to explain the behavior and properties

of light which couldn’t be explained by the particle theory. This theory suggested that light could

also be diffracted and produce interference effects.

De Borglie suggested that matter could also exhibit a dual nature. He proposed that there was a

relationship between the momentum of any particle and its wavelength.

Further research by Davission and Germer confirmed this relationship when they were able to

succeed in diffracting electrons.

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Production of X-Rays

What are X- Rays?

X- Rays are the electromagnetic waves with short wavelengths, usually 10-10m or less.

How are they produced?

Diagram Shows a Coolidge Type X-Ray Tube

The filament is a very hot wire (usually made of tungsten). When this wire is heated, the kinetic

energy of the electrons in the wire also increases causing them to vibrate more quickly and

vigorously until they reach a point where they are able to escape the wire.

The electrons from the filament travel at high speeds and collide with the target.

The target (anode) is set a large positive potential (with reference to the filament lamp) by using

a high voltage unit. By doing this the negatively charged electrons experience a strong pull from

the positive target (anode) and the electrons accelerate towards the target with high speeds. This

is aided by the evacuation of the tube (the vacuum).

Upon collision with the target, the electrons are decelerated (lose speed and hence lose energy).

This type of energy loss is what produces the X- rays. The rest of the energy which was not

“lost” (usually 99%) goes towards the production of heat.

Since 99% of the energy which the electrons possess is converted into heat energy, the tube is

designed with a cooling system to stop it from melting. In the diagram above this is shown by the

cooled copper rod which conducts the emitted heat by the fast moving electrons away from the

target. The rod is cooled by circulating oil through it or by using cooling fins.

X- rays are produced with the interaction of the electrons with the atoms of the target material

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X-rays produced by the Interaction with Atoms of Target Material

The electrons which bombard the target are very energetic and are capable of knocking electrons

out of the deep lying energy levels of that atom. They are able to do this because the electrons in

the lower energy levels are quite stable and do not move around as the one in the higher level do

and are easier to target.

When one of the electrons from the lower energy is knocked out it leaves a space or a hole. The

electrons in the higher level are seeking to become more stable and seek to occupy the space left

by the electrons which was knock out. The electron from the higher level “falls” into the space.

In order to do this, it must lose some of its energy since all the electrons at a particular level must

have the same energy. The energy at which that electron loses in order to occupy that stable

place in a lower level goes towards the production of X-rays.

When an electrons moves from a level with high energy (E2) to one of lower energy (E1), the

frequency (f) of the emitted radiation is calculated by:

𝐸2 − 𝐸1 = ℎ𝑓

where h = Planck’s constant (6.626 × 10-34 Js)

(N.B. the more energy emitted means an increase in frequency but this also means a decrease in

wavelengths since f α 1/ʎ).

Energy Levels

The energies of electrons in atoms can have only specific values which are called the energy

levels of the atoms. All the atoms of a named element have the same set of energy levels which

gives the characteristics of the atom and they differ from every other element.

These energy levels are expressed in electronvolts and are usually represented as a series of

horizontal lines. When an electron occupies the lowest level in the atom it is said to be in its

ground state. If it absorbs energy, the atom may be promoted into a higher energy level. The

atom is unstable and is said to be in an excited state.

After a short and random interval the electron ‘falls’ back into the lowest level so that atom can

return to its ground state. The energy that was originally absorbed is emitted as electromagnetic

waves.

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Each energy level is described by a quantum number (n). The lowest level is n = 1, then n = 2,

etc. The highest energy level is n = ∞, and at this level the energy is zero. When an electron

reaches this level, the electron becomes free of the atom and the atom is said to be ionized,

Energy Levels of

Hydrogen atom

X-Ray Emission Spectra

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The diagram above shows a typical X-ray spectrum which has two distinct components:

(i) a background of continuous radiation, the minimum wavelength of which depends on

the operating voltage of the tube. (the energy of the electrons bombarding the target)

(ii) very intense emission at few discrete wavelengths (an X-ray spectrum). These

wavelengths are characteristic of the target material and are independent of the

operating voltage.

The Continuous Background

This is produced by electrons colliding with the target and being decelerated. It is called

continuous because there is large range of energy due to the fact that the electrons do not lose

any one specific value of energy. The energy that each electron lose may be different from other

electrons because of the speed that the electron was originally travelling or the angle at which the

electron collide with the target. All of the factors mention determines what fraction of the

electron’s original energy value will be lost resulting in the large range of values.

The Line Spectrum

This is produced by the electrons knocking out a stable electron from the atom of the target

material. It can be seen that specific values of energy needs to be lost in order for the higher level

electrons to fill the space in the lower level. Therefore a particular energy value needs to be lost.

The total X- ray product is a combination of that produced by the continuous spectrum and the

line spectrum.

Properties of X- Rays

i. They travel in straight lines at the velocity at the speed of light

ii. They cannot be deflected by electric of magnetic fields hence they are not charged

particles

iii. They penetrate matter. Penetrating is the least with materials of high density

iv. They can be reflected but only at large angles of incidences

v. They can be diffracted

vi. The intensity of the X-ray increases with the number of electrons hitting the target and

the filament current. It is also increased by increasing the PD across the tubes.

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vii. The penetrating power of an X-ray increase with the PD across the tube. An X-ray with a

low penetrating power is called a soft X-ray while one with a high penetrating power is

called a hard X-ray.

viii. The most energetic X-rays (those whose wave lengths are ʎmin) are the result of the

bombarding losing all their energy at once. Since the energy of the electrons depends on

the operating voltage, so does ʎmin. Hence the higher the voltage, the smaller the value of

ʎmin.

𝜆𝑚𝑖𝑛 = ℎ𝑐

𝑒𝑉

where h - Planck’s constant

c - speed light

e - electron charge

V - tube voltage

Addition Properties (used to detect X-rays)

i. They ionize gases through which they pass

ii. They blacken photographic film

iii. They can produce fluorescence

iv. They can produce photo-electric emission

Uses of X-Rays

i. Used in medicine to locate bone fractures and to destroy cancer cells. (Bones absorb X-

rays more than flesh)

ii. To detect cracks in metals

iii. To investigate the structure of crystals

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X- Ray Absorption Spectra

X-Rays passing through an Absorber

We can calculate the emerging intensity (I1) of a beam of X-ray of original intensity (I0) after it

has passed through a slab of material of thickness (x1) and absorption coefficient (µ1) by using

the equation

𝐼1 = 𝐼0𝑒−𝜇1𝑥1

It follows from the equation that

ln 𝐼1 = ln(𝐼0𝑒−𝜇1𝑥1)

ln 𝐼1 = ln(𝐼0) + ln( 𝑒−𝜇1𝑥1)

∴ ln 𝐼1 = −𝜇1𝑥1 + ln(𝐼0)

where y = ln I1, x = x1, m = -µ1, c = ln (I0)

If a second slab of thickness (x2) and absorption coefficient (µ2) is placed directly behind the first

slab. We can calculate the value for the emerging intensity (I2) of the beam by using

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𝐼1 = 𝐼0𝑒−𝜇1𝑥1 and 𝐼2 = 𝐼1𝑒−𝜇2𝑥2

Hence 𝐼2 = 𝐼0𝑒−𝜇1𝑥1 × 𝑒−𝜇2𝑥2

𝐼2 = 𝐼0𝑒(−𝜇1𝑥1)+( −𝜇2𝑥2)

∴ 𝐼2 = 𝐼0𝑒−(𝜇1𝑥1+ 𝜇2𝑥2)

We can use a graph to represent the way that µ varies with the wavelength (ʎ) for any absorber.

As ʎ increases, µ and the absorption rapidly increases until ʎ = ʎk. This point is called K

absorption edge. Up to this point, absorption is mainly due to the ejections of electrons from the

K shell of the absorber.

The sudden drop in µ at the point ʎk occurs because the X-rays of longer wavelengths do not

have enough energy to eject K shell electrons. As ʎ increase beyond ʎk, the absorption increases

again and is due mainly to the ejection of electrons from the higher lying L shell. The L shell

consists of three energy levels (LI, LII and LIII) which are very close to each other.

The Difference between an Emission and Absorption Spectrum

Emission spectrum is a range of radiation produced by the change in energy level of electrons

which results in the energy being emitted, but the absorption spectrum is caused by an object

absorbing radiation of a particular wavelength from an incident beam and leaving dark bands or

darks spaces in the emergent beam. Absorption emission spectra can be used to determine the

materials because certain materials absorb and emit specific wavelengths of radiation.

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De Broglie’s Equation

In 1924 Louis de Broglie presented a thesis stating that matter possesses wave-particle duality

nature. He proposed that any particle of momentum (p) has an associated wavelength (ʎ) called

the de Broglie wavelength.

The de Broglie Equation states that

𝜆 = ℎ

𝜌=

ℎ

𝑚𝑣

∴ 𝜌 = ℎ

𝜆

where m - relative mass

v - velocity of particle

h - Planck’s constant

p - momentum of particles

Table of Electromagnetic Waves showing Frequency and Wave Length Ranges

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The Structure of the Atom

The Atom

Scientists such as Rutherford, Geiger, Thomson, Mardsen, Bohr and Chadwick helped

established modern views of the atom.

Before 1897

Atoms were thought to be small invisible particles

1902 -Thomson

Thomson showed that negative charges called electrons existed within matter. He suggested that

the atom resembled a plum pudding where the electrons were represented by the seeds and the

body of the plum pudding was the positive charge.

1906 - Rutherford

During this time Rutherford observed that alpha particles passing through a thin sheet of mica

without making holes in it.

1911- Geiger and Marsden

A narrow beam of alpha particles from a radon sourced was fired at a thin metal foil. A glass

screen coated with zinc sulphide was used to detect the scattered alpha particles. The experiment

was carried out in a darkened room under a microscope and whenever an alpha particle strike the

screen a faint flash of light was observed.

It was found that the majority of the alpha particles deviated through small angles and that only a

small number deviated more than 900 and even less deflected back towards the source. These

observations apposed Thomson’s ‘plum pudding’ model of the atom.

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From this experiment it was therefore concluded that the atom has a small positively charged

core which contained most of the mass of the atom and which was surrounded by orbiting

electrons.

1913 - Rutherford and Bohr

These two scientists came up with a new model that suggested that the atom had a dense central

core, which was positive, and a very small negative part compared to the rest of the atom. They

suggested that the most of the volume occupied by the atom was an empty space and negative

electrons orbited around the nucleus

1932 - Chadwick

Chadwick identified the neutron, a neutral particle found within the nucleus of the atom.

Chadwick’s experiment supported the new model proposed by Rutherford and Bohr.

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Today

We still picture the atom today to have a very small and dense nucleus that consists of protons

and neutrons, surrounding by a cloud of negative electrons. The electrons cannot be pinpointed

since it is thought to behave as a wave and a particle.

Atomic Structure Atoms are composed of neutrons, protons and electrons

Standard Notation

𝑋𝑍𝐴

where X is the element symbol

Z is the atomic number (proton number)

A is the mass number (nucleon number)

PARTICLE CHARGE MASS

PROTON + 1 1

ELECTRON - 1 1/ 1840

NEUTRON None 1

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Atomic Number (Z)

The atomic number of an element is the number of protons found in the nucleus of that element.

E.g. Carbon has an atomic number of 6; hence there are 6 protons in its nucleus.

Mass Number (A)

The mass number of an element is the number of protons plus the number of neutrons.

(A = Z + N)

Atoms are neutral particles and have the same number of protons as electrons.

E.g. Sodium has 11 protons; therefore it must also have 11 electrons.

Examples:

PARTICLE NO. OF

PROTONS

NO. OF

NEUTRONS

ATOMIC

NO.

MASS NO.

𝐶𝑙1737 17 20 17 37

𝐶612 6 6 6 12

𝐺𝑎3170 31 38 31 70

𝑃𝑜84210 84 126 84 210

Isotopes

Isotopes are different atoms of the same element. They have the same atomic number but

different mass numbers. Isotopes of the same elements have the same number of protons and

electrons but different neutrons.

Examples

𝑯𝟏𝟏 𝑯𝟏

𝟐 𝑯𝟏𝟑

𝐶612 𝐶6

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Millikan’s Determination of Electron Charge (e)

The principle of Millikan’s experiment is to measure the terminal velocity of a small charged

drop of oil under gravity; then to oppose its motion with an electric field in such a manner that it

remains stationary.

Terminal velocity: An object falling air experiences a force which opposes the motion of the

object (viscous drag). The initial downward force is greater than the drag forces and the object

accelerates. However, as the drag forces increase with the velocity of the object, the gravitational

force and the drag forces eventually become equal. When this occurs there is no more

acceleration. The object has reached its maximum speed and this is called terminal velocity.

When there is no electric field: the forces acting on the oil drop are shown in the diagram

above. Once the oils drop has reached the terminal velocity, there is no more acceleration and

therefore we can state that

𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑢𝑝𝑡ℎ𝑟𝑢𝑠𝑡 𝑑𝑢𝑒 𝑡𝑜 𝑎𝑖𝑟 + 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑑𝑟𝑎𝑔 … ①

We know that the weight of the drop of oil can be calculated by

𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑑𝑟𝑜𝑝 × 𝑑𝑒𝑛𝑖𝑠𝑡𝑦 𝑜𝑓 𝑜𝑖𝑙 × 𝑔𝑟𝑎𝑣𝑖𝑡𝑦

Therefore 𝑤𝑒𝑖𝑔ℎ𝑡 = 4

3𝜋𝑟3𝜌0𝑔 …②

where r = radius of the oil drop

p0 = density of the oil drop

Also 𝑢𝑝𝑡ℎ𝑟𝑢𝑠𝑡 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑎𝑖𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 𝑏𝑦 𝑜𝑖𝑙 𝑑𝑟𝑜𝑝

= 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑜𝑖𝑙 𝑑𝑟𝑜𝑝 × 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑎𝑖𝑟 × 𝑔𝑟𝑎𝑣𝑖𝑡𝑦

∴ 𝑢𝑝𝑡ℎ𝑟𝑢𝑠𝑡 = 4

3𝜋𝑟3𝜌𝑎𝑔 … ③

where pa = density of the oil drop

According to Stokes’ Law, we can calculate the viscous drag by using the equation

𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑑𝑟𝑎𝑔 = 6𝜋𝑟𝜂𝑣 … ④

where ƞ = coefficient of viscosity of air

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Substituting ②, ③ and ④ into ①; we get that

4

3𝜋𝑟3𝜌0𝑔 =

4

3𝜋𝑟3𝜌𝑎𝑔 + 6𝜋𝑟𝜂𝑣 … ⑤

When an electric field is applied such that the oil drop is stationary: the forces acting on the

oil drop (as shown in the diagram above) are such that the oil drop has no velocity and no

acceleration.

We can therefore state the equation

𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑢𝑝𝑡ℎ𝑟𝑢𝑠𝑡 + 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑓𝑜𝑟𝑐𝑒

We can state

4

3𝜋𝑟3𝜌0𝑔 =

4

3𝜋𝑟3𝜌𝑎𝑔 + 𝑄𝐸 … ⑥

where Q = the charge of the oil drop

E = electric field strength

When we subtract ⑤ from ⑥, we get that

0 = 𝑄𝐸 − 6𝜋𝑟𝜂𝑣

∴ 𝑄 = 6𝜋𝑟𝜂𝑣

𝐸 … ⑦

However, Millikan measured E and v and did and did another experiment to find ƞ. He was not

able to measure r directly but rearranging ⑤ to get that

4

3𝜋𝑟3(𝜌0 − 𝜌𝑎)𝑔 = 6𝜋𝑟𝜂𝑣

∴ 𝑟 = (9 𝜂𝑣

2(𝜌0−𝜌𝑎)𝑔)

1

2 … ⑧

Substituting ⑧ into ⑦, we get that

𝑄 = 6𝜋𝜂

𝐸(

9 𝜂𝑣

2(𝜌0 − 𝜌𝑎)𝑔)

12 𝑣

But since the density of air at RTP is extremely small compared to a drop of oil, we can use the

equation

𝑄 = 6𝜋𝜂

𝐸(

9 𝜂𝑣

2𝑝0𝑔

)

12 𝑣

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NUCLEAR STABILITY, FUSION AND FISSION

Einstein’s Mass – Energy Equation

Einstein stated that the relationship between the mass (m) and the speed of light (c) is equivalent

to the amount of Energy (E), and can be calculated by the equation

𝐸 = 𝑚𝑐2

Therefore as there is a change in energy there is also a change in mass.

The unified atomic mass unit (u) is defined as 1

12 the mass of the Carbon – 12 atom

Mass of 𝐶612 = 12𝑔

So 12

6 ×1023 𝑔 = 12

6 ×1026 𝑘𝑔 = 12 𝑢

12 u is the mass of one particle in 1 mole of Carbon 12 atoms, therefore

1𝑢 = 12

12 × (6.0 × 1026)

∴ 1 𝑢 = 1.661 × 10−27 𝑘𝑔

And we can state that

1 𝑢 = 932 𝑀𝑒𝑉

Mass Defect

The mass of nucleus is always less than the total mass of it individual nucleons. The reduction in

mass occurs because combining the nucleons causes some of their mass to be released as energy

(in the form of γ – rays). The difference in mass is called the mass defect.

𝑚𝑎𝑠𝑠 𝑑𝑒𝑓𝑒𝑐𝑡 = 𝑚𝑎𝑠𝑠 𝑛𝑢𝑐𝑙𝑒𝑜𝑛𝑠 − 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑛𝑢𝑐𝑙𝑒𝑢𝑠

or

𝑚𝑎𝑠𝑠 𝑑𝑒𝑓𝑒𝑐𝑡 = 𝑜𝑓 𝑛𝑢𝑐𝑙𝑒𝑜𝑛𝑠 𝑎𝑛𝑑 𝑒𝑙𝑐𝑡𝑟𝑜𝑛𝑠 − 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑎𝑡𝑜𝑚

Binding Energy and Nuclear Forces

The nucleus consists of protons and neutrons. The protons repel each other, yet the nucleons are

held together in the nucleus. In order for the nucleons to be stable there must be strong force

between them. This force is called the nuclear force.

Any attempt to separate the nucleons requires energy. When the nucleons come together in the

nucleus, there is loss of energy which is equal to the binding energy. The binding energy,

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therefore is the energy required to completely split up the nucleons into its neutrons, protons and

electrons which exists in the nucleus of the atom.

𝑏𝑖𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑒𝑟𝑔𝑦 (𝐽) = 𝑚𝑎𝑠𝑠 𝑑𝑒𝑓𝑒𝑐𝑡 (𝑘𝑔) × 𝑐 2 (𝑚𝑠−1)2

or

𝑏𝑖𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑒𝑟𝑔𝑦 (𝑀𝑒𝑉) = 932 × 𝑚𝑎𝑠𝑠 𝑑𝑒𝑓𝑒𝑐𝑡 (𝑢)

Worked Example

Calculate the mass defect (in both u and kg) and the r=energy released (in both J and MeV)

𝐿𝑖37 + 𝐻1

1 → 𝐻𝑒24 + 𝐻𝑒2

4 + 𝑄

Atomic masses: Li = 7.016 u H = 1.008 u He = 4.004 u

𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 = ( 7.016 + 1.008) = 8.024 𝑢

𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑎𝑓𝑡𝑒𝑟 = 2(4.004) = 8.008 𝑢

𝑚𝑎𝑠𝑠 𝑑𝑒𝑓𝑒𝑐𝑡 (𝑖𝑛 𝑢) = (8.024 − 8.008) = 0.016 𝑢

𝑚𝑎𝑠𝑠 𝑑𝑒𝑓𝑒𝑐𝑡 (𝑖𝑛 𝑘𝑔) = 0.016 × (1.66 × 10−27) = 2.66 × 10−29 𝑘𝑔

𝑈𝑠𝑖𝑛𝑔 𝐸 = 𝑚𝑐2

𝐸 (𝑖𝑛 𝐽𝑜𝑢𝑙𝑒𝑠) = (2.66 × 10−29) × (3.0 × 108)2 = 2.39 × 10−12𝐽

𝐸 (𝑖𝑛 𝑀𝑒𝑉) = 2.39 × 10−12

1.6 × 10−19 = 1.49 × 107𝑒𝑉 = 14.9 𝑀𝑒𝑉

Variation of Binding Energy per Nucleon with Mass Number

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Nuclear Fission

Nuclear Fission is the disintegration of a heavy nucleus of a heavy nucleus into two lighter

nuclei. The energy is released because the average binding energy per nucleon of the fission

products is greater than the average binding energy per nucleon of the fission products is greater

than that of the parent.

Example of a fission reaction

𝑈92235 + 𝑛0

1 → 𝐵𝑎56141 + 𝐾𝑟36

92 + 3 𝑛01

Worked Example

Nuclear Fusion

Nuclear Fusion is the combination of two light nuclei to produce a heavier nucleus. The energy

released by this process is less than that which results from fission.

Example of fusion reaction

𝐻12 + 𝐻1

2 → 𝐻𝑒23 + 𝑛0

1

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Worked Example

The Relationship between Binding Energy per Nucleon and Nuclear Fusion and Fission

A nucleus becomes more stable when it releases energy. Therefore the greater the stability the

more energy is needed to break the nucleus up. So the higher the binding energy, the more stable

the element. (mass defect is directly proportional to the binding energy)

Since nuclear reactions occur in order to increase the stability of the nucleus; fusion and fission

reactions will occur in order to produce a nucleus with a higher binding energy.

Note that the rising part of the binding energy curve indicates that elements with lower mass

numbers can produce energy by fusion. The falling portion of the curve indicates that the heavy

elements can produce energy by fission.

Note that Iron does not usually undergo nuclear reactions because it is very stable, hence it has

the highest binding energy.

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RADIOACTIVITY

Radioactivity is the spontaneous emissions from the nucleus of certain atoms, of either alpha,

beta or gamma radiation. These radiations are emitted when the nuclei of the radioactive

substance breaks down to form a new and more stable nuclei.

One of the early workers with this phenomenon was Marie Curie. She experimented with

uranium compounds and discovered new elements such as polonium and radium. These

substances emitted invisible radiation that eventually killed her in 1934 from overexposure.

Nature of Radioactive Emissions

Alpha particles () are helium nuclei. They have 2 protons and 2 neutrons existing together.

Beta particles () are electrons moving at high speeds.

Gamma rays () are electromagnetic waves of very short wavelengths.

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Cloud Chamber

Radioactivity can be detected by using by using a device called the diffusion cloud chamber. The

device consists of a cylinder chamber under which dry ice is placed. Inside the chamber is filled

with alcohol vapour, which is cooled about –65 oC by using dry ice. When the radioactive source

is placed close to the base, the inside is illuminated with white vapour trails, which can be seen

shooting from the source. The vapour trails formed, shows the paths taken by the radiation and

can be used to distinguish the type of radiation present.

The Effect of Magnetic Fields on Radioactive Emissions

Some radioactive emissions are deflected by both electric and magnetic fields; the diagram

below shows the effects of magnetic fields on particles, particles and rays.

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Gamma () rays are not affected by magnetic fields since they have no charge, hence they do not

deflect.

Alpha () particles show a slight deviation to the left as shown in the diagram above. By using

Fleming Left Hand Rule we can determine that the alpha particles are positively charged.

Beta () particles deviate to the right, which shows that they have a negative charge.

Note that alpha particles deviate less than beta particles because they are massive compared to

beta particles.

Radioactive Decay

If an isotope is radioactive it has an unstable arrangement in its nuclei. The emission of alpha or

beta particles can make the isotope more stable. This type of reaction however alters the number

of protons or neutrons in the nucleus making the nucleus of a different element. The original

nucleus is called the parent nucleus and the new nucleus is called the daughter nucleus. (N.B.

Further decay can also produce a granddaughter nucleus.) The daughter nucleus and the

emitting products are called decay products.

Radioactive decay, which occurs naturally, is random and spontaneous and there is no way of

telling how much of the nuclei will decay or when they will decay.

Alpha Particle Decay

When a nucleus decays by emitting alpha () particles, its atomic mass number (Z) decreases by

two and its mass number (A) decreases by four. This type of decay occurs when the nuclei has

too many protons to be stable.

General Equation:

𝑋𝑍𝐴 → 𝑌𝑍−2

𝐴−4 + 𝐻𝑒24 + 𝐸𝑛𝑒𝑟𝑔𝑦

Examples

𝑅𝑎88226 → 𝑅𝑛86

222 + 𝛼24 + 𝐸𝑛𝑒𝑟𝑔𝑦

𝑃𝑜84212 → 𝑃𝑏82

208 + 𝐻𝑒24 + 𝐸𝑛𝑒𝑟𝑔𝑦

𝑈92238 → 𝑇ℎ90

234 + 𝐻𝑒24 + 𝐸𝑛𝑒𝑟𝑔𝑦

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Beta Particle

Beta emission causes the proton number (Z) to increase by 1, but the nucleon number (A) does

not change. Beta particles are emitted when the nuclei have too many neutrons to be stable.

General Equation:

𝑋𝑍𝐴 → 𝑌𝑍+1

𝐴 + 𝑒−10 + 𝐸𝑛𝑒𝑟𝑔𝑦

Examples

𝐶614 → 𝑁7

14 + 𝛽−10 + 𝐸𝑛𝑒𝑟𝑔𝑦

𝑃𝑏82212 → 𝐵𝑖83

212 + 𝑒−10 + 𝐸𝑛𝑒𝑟𝑔𝑦

𝐴𝑐89230 → 𝑇ℎ90

230 + 𝛽−10 + 𝐸𝑛𝑒𝑟𝑔𝑦

Gamma Ray Emissions

With some isotopes the emission of alpha particles and/ or beta particles leaves the neutrons and

protons in an excited arrangement. As the protons and neutrons arrange themselves into a more

stable arrangement they lose energy. This energy is emitted as a burst of gamma radiation.

Gamma radiation does not cause changes in the mass number (A) or the atomic number (Z) of

the isotope.

General Equation:

𝑋𝑍𝐴 → 𝑌𝑍

𝐴 + 𝛾 𝑟𝑎𝑦𝑠

Examples

𝑃𝑏82212 → 𝑃𝑏82

212 + 𝛾 𝑟𝑎𝑦𝑠

𝑁714 → 𝑁7

14 + 𝛾 𝑟𝑎𝑦𝑠

Exponential Law of Radioactive Decay

This law states that for large numbers of any particular nuclei the rate of decay is proportional to

the number of parent nuclei. 𝑑𝑁

𝑑𝑡= −𝜆𝑁 … ①

where λ = positive constant of proportionality called the decay constant (unit: s-1)

−𝑑𝑁

𝑑𝑡 = the rate of decay and is called the activity of the source. (unit: becquerel, Bq)

NB: 1 Bq = 1 disintegration per second and 1 Ci (curie) = 3.7 ×1010 s-1

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Rearranging ① we get that

𝑑𝑁

𝑁= −𝜆 𝑑𝑡 … ②

Integrating ② we get that

∫𝑑𝑁

𝑁= ∫ −𝜆 𝑑𝑡

∫1

𝑁 𝑑𝑁 = −𝜆 ∫ 𝑑𝑡 … ③

Since we know that ∫1

𝑥 = ln 𝑥 and ∫ 𝑑𝑡 = 𝑡, then ③ becomes

ln 𝑁 = − 𝜆𝑡 + 𝑐 … ④

Given that N0 = the number of original radioactive atoms (i.e. at t = 0), and substituting into ④,

ln 𝑁0 = − 𝜆 (0) + 𝑐

∴ 𝑐 = ln 𝑁0 … ⑤

Substituting ⑤ into ④, we get that

ln 𝑁 = − 𝜆𝑡 + ln 𝑁0

ln 𝑁 − ln 𝑁0 = − 𝜆𝑡

ln (𝑁

𝑁0) = − 𝜆𝑡

∴ 𝑁

𝑁0= 𝑒− 𝜆𝑡

Hence, we can conclude that

𝑁 = 𝑁0𝑒− 𝜆𝑡 … ⑥

Since activity is proportional to the number of parent nuclei, we can also say that

𝐴 = 𝐴0𝑒− 𝜆𝑡 … ⑦

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Graph showing the Exponential Nature of Radioactive Decay

If a radioactive source has a short half-life, apparatus can be used to obtain data to plot a decay

curve. It is not easy to measure the number of nuclei in a sample of radioactive material but the

rate of decay can be used to indicate how much of the nuclei is present.

To plot a radioactive decay curve we need to find the value for the background count, this would

be subtracted from each reading obtained. Readings are then taken at regular intervals and the

corrected count rate is plotted against the time. We can then use the graph to find the half-life of

the substance.

Activity (A)

This is the number of particle emission per second from a radioactive source and hence is simply

the rate of decay. (unit: Becquerel, Bq)

Decay Constant (λ)

This is the positive constant of proportionality. Decay is the change of a property such that the

value decreases with time

𝐴 = 𝜆𝑁 where A = activity

λ = decay constant

N = number of parent nuclei

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Half-Life (T1/2)

The half-life of a radioactive substance is the time taken for half of the unstable atoms to decay

(decrease by half of its original mass).

If we represented the half -life by t = T1/2 and N = N0 / 2 and substituted these values into ⑥, we get that

𝑁0

2= 𝑁0𝑒− 𝜆𝑇1/2

1

2= 𝑒− 𝜆𝑇1/2

ln (1

2) = − 𝜆𝑇1/2

−0.6931 = − 𝜆𝑇1/2

∴ 𝑇1/2 = 0.6931

𝜆 𝑜𝑟 𝑇1/2 =

ln 2

𝜆

N.B. Radioactive decay occurs randomly, hence there is no way of predicting when a particular

nucleus will degrade. The radioactive process is not affected by temperature change, pressure

change or chemical change.

Uses of Radioactivity

Cancer cells can be destroyed by γ – radiation from high – activity source of Cobalt-60

The thickness of metal sheets can be monitored by using γ –rays and a detector. Thicker

sheets absorb more γ- rays.

A small quantity of short-lived radioactive liquid can be used to detect the exact positions

of pipes and leaks.

Radioactive dating (example Carbon - 14 dating)

Radioisotopes can be used to monitor different abilities in both plants and humans.

Operations of Simple Detectors

(See Muncaster A’Level Physics Page 831 – 836 – Detectors of Radiation)

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Worked Example 1

Worked Example 2

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