# Quantum Physics and Nuclear Physics

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Quantum Physics and Nuclear Physics

Quantum Physics Revision
When an electron falls to a lower energy level, the change in energy (ΔE) is emitted as a photon of e-m radiation. ΔE = E2 – E1 The change in energy is proportional to the frequency of the emitted photon: ( E = Photon energy h = Plank’s constant = 6.6 x m2kgs-1 ) E = hf

MAX PLANCK PHOTOELECTRIC EFFECT John Parkinson

The photoelectric effect
Quantum Phenomena Breithaupt pages 30 to 43 The photoelectric effect The photoelectric effect is the emission of electrons from the surface of a material due to the exposure of the material to electromagnetic radiation. For example zinc emits electrons when exposed to ultraviolet radiation. If the zinc was initially negatively charged and placed on a gold leaf electroscope, the electroscope’s gold leaf would be initially deflected. However, when exposed to the uv radiation the zinc loses electrons and therefore negative charge. This causes the gold-leaf to fall. - - .- Gold Leaf Electroscope April 8th, 2010

At the end of the nineteenth century, Classical Electromagnetic Wave Theory thought of light waves as being like water waves. A The wave’s Intensity or energy was directly proportional to the square of the Amplitude, A. John Parkinson

Potassium metal undergoes photoemission with blue and green light, but not with red light.
Blue light Emission! Emission! e Green light Red light Nothing!! potassium metal John Parkinson

THE CLASSICAL THEORY SUGGESTS TRYING MORE INTENSE LIGHT
Nothing!! Nothing!! potassium metal John Parkinson

The Classical Theory must be wrong!!!!!
John Parkinson

Limitation of Wave Theory of Light
Threshold Frequency Electrons will only be emitted from zinc by photoelectric emission if the electromagnetic radiation incident upon its surface has a frequency of 1 x 1015 Hz or above. This is called the threshold frequency of zinc. Limitation of Wave Theory of Light Wave theory would suggest that once enough visible light energy had been absorbed by the zinc, the electron would be able to escape. This is not the case. No matter how intense the incident radiation, if its frequency is below the threshold frequency for a particular material, no photoelectric emission will occur.

Experimental observations
Quantum Phenomena Breithaupt pages 30 to 43 Experimental observations Threshold frequency The photoelectric effect only occurs if the frequency of the electromagnetic radiation is above a certain threshold value, f0 Variation of threshold frequency The threshold frequency varied with different materials. Affect of radiation intensity The greater the intensity the greater the number of electrons emitted, but only if the radiation was above the threshold frequency. Time of emission Electrons were emitted as soon as the material was exposed. Maximum kinetic energy of photoelectrons This depends only on the frequency of the electromagnetic radiation and the material exposed, not on its intensity. INTENSITY OF RADIATION DOES NOT ATTECT ENERGY OF EMITTED ELECTRONS April 8th, 2010

Problems with the wave theory
Quantum Phenomena Breithaupt pages 30 to 43 Problems with the wave theory Up to the time the photoelectric effect was first investigated it was believed that electromagnetic radiation behaved like normal waves. The wave theory could not be used to explain the observations of the photoelectric effect – in particular wave theory predicted: that there would not be any threshold frequency – all frequencies of radiation should eventually cause electron emission that increasing intensity would increase the rate of emission at all frequencies – not just those above a certain minimum frequency that emission would not take place immediately upon exposure – the weaker radiations would take longer to produce electrons. April 8th, 2010

Limitation of Wave Theory of Light
Threshold Frequency Electrons will only be emitted from zinc by photoelectric emission if the electromagnetic radiation incident upon its surface has a frequency of 1 x 1015 Hz or above. This is called the threshold frequency of zinc. Limitation of Wave Theory of Light Wave theory would suggest that once enough visible light energy had been absorbed by the zinc, the electron would be able to escape. This is not the case. No matter how intense the incident radiation, if its frequency is below the threshold frequency for a particular material, no photoelectric emission will occur.

Photons – the Quantum Model
In 1900 Max Planck came up with the idea of energy being ‘quantised’ in some situations. i.e. existing in small ‘packets’. In 1905 Einstein suggested that all e-m radiation is emitted in small quanta called photons rather than in a steady wave. Intensity of radiation depends on the number of photons being emitted per second (not amplitude as suggested by the wave model). Energy per photon depends upon its frequency: E = hf

Einstein’s explanation
Quantum Phenomena Breithaupt pages 30 to 43 Einstein’s explanation Electromagnetic radiation consisted of packets or quanta of energy called photons The energy of these photons: depended on the frequency of the radiation only was proportional to this frequency Photons interact one-to-one with electrons in the material If the photon energy was above a certain minimum amount (depending on the material) the electron was emitted any excess energy was available for electron kinetic energy Einstein won his only Nobel Prize in 1921 for this explanation. This explanation also began the field of Physics called Quantum Theory, an attempt to explain the behaviour of very small (sub-atomic) particles. April 8th, 2010

The electron can then either
Quantum Theory of the Photoelectric Effect The photons are sufficiently localized, so that the whole quantum of energy [ hf ] can be absorbed by a single electron at one time. The electron can then either share its excess energy with other electrons and the ion lattice or it can use the excess energy to fly out of the metal. Because of the interaction of this electron with other atoms, it requires a certain minimum energy to escape from the surface. The minimum energy required to escape depends on the metal and is called the work function, Φ. John Parkinson

For electron emission, the photon's energy has to be greater than the work function .
The maximum kinetic energy the released electron can have is given by: EK = photon energy – the work function. EK = hf - Φ For every metal there is a threshold frequency, f0, where hf0 = Φ , that gives the photon enough energy to produce photoemission. The number of photoelectrons emerging from the metal surface per unit time is proportional to the number of photons striking the surface that in turn depends on the intensity of the incident radiation It follows that the photo electric current is proportional to the intensity of the radiation provided the frequency of radiation is above threshold frequency. John Parkinson

Quantum Theory of the Photoelectric Effect
In 1905 Einstein developed Planck’s idea, that energy was quantised in quanta or photons, in order to explain the photoelectric effect. Electromagnetic radiation is emitted in bursts of energy – photons. The energy of a photon is given by E = hf, where f is the frequency of the radiation and h is Planck’s constant. [h = 6.6 x Js] But velocity of light = frequency times wavelength Substituting into E = hf John Parkinson

BLUE !!! λ frequency the visible spectrum violet light light 400 nm
red light light 700 nm uv light < 400 nm λ Blue photon Red photon Which photon has the most energy ????? BLUE !!! John Parkinson

Photon energy (revision)
Quantum Phenomena Breithaupt pages 30 to 43 Photon energy (revision) photon energy (E) = h x f where h = the Planck constant = 6.63 x Js also as f = c / λ; E = hc / λ Calculate the energy of a photon of ultraviolet light (f = 9.0 x 1014 Hz) (h = 6.63 x Js) E = h f = (6.63 x Js) x (9.0 x 1014 Hz) = 5.37 x J April 8th, 2010

The photoelectric equation
Quantum Phenomena Breithaupt pages 30 to 43 The photoelectric equation hf = φ + EKmax where: hf = energy of the photons of electromagnetic radiation φ = work function of the exposed material EKmax = maximum kinetic energy of the photoelectrons Work function, φ This is the minimum energy required for an electron to escape from the surface of a material April 8th, 2010

Threshold frequency f0 As: hf = φ + EKmax
Quantum Phenomena Breithaupt pages 30 to 43 Threshold frequency f0 As: hf = φ + EKmax If the incoming photons are of the threshold frequency f0, the electrons will have the minimum energy required for emission and EKmax will be zero therefore: hf0 = φ and so: f0 = φ / h April 8th, 2010

Question 1 f0 = φ / h = (1.2 x 10-19 J) / (6.63 x 10-34 Js)
Quantum Phenomena Breithaupt pages 30 to 43 Question 1 Calculate the threshold frequency of a metal if the metal’s work function is 1.2 x J. (h = 6.63 x Js) f0 = φ / h = (1.2 x J) / (6.63 x Js) threshold frequency = 1.81 x 1014 Hz April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Question 2 Calculate the maximum kinetic energy of the photoelectrons emitted from a metal of work function 1.5 x J when exposed with photons of frequency 3.0 x 1014 Hz. (h = 6.63 x Js) hf = φ + EKmax (6.63 x Js) x (3.0 x 1014 Hz) = (1.5 x J) + EKmax EKmax = x x 10-19 = x J maximum kinetic energy = 4.89 x J April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 The vacuum photocell The photocell is an application of the photoelectric effect Light is incident on a metal plate called the photocathode. If the light’s frequency is above the metal’s threshold frequency electrons are emitted. These electrons passing across the vacuum to the anode constitute and electric current which can be measured by the microammeter. April 8th, 2010

The electromagnetic radiation releases electrons from the metal cathode. These electrons are attracted to the anode and complete a circuit allowing a current to flow Radiation mA Anode +ve vacuum Cathode -ve electrons John Parkinson

If the polarity is reversed, the pd across the tube can be increased until even the most energetic electrons fail to cross the tube to A. The milliammeter then reads zero. Radiation mA A C electrons electrons V The p.d. across the tube measures the maximum kinetic energy of the ejected electrons in electron volts. John Parkinson

Obtaining Planck’s constant
Quantum Phenomena Breithaupt pages 30 to 43 Obtaining Planck’s constant By attaching a variable voltage power supply it is possible to measure the maximum kinetic energy of the photoelectrons produced in the photocell. The graph opposite shows how this energy varies with photon frequency. hf = φ + EKmax becomes: EKmax = hf – φ which has the form y = mx +c with gradient, m = h Hence Planck’s constant can be found. April 8th, 2010

Maximum EK emitted electrons / J
metal A Maximum EK emitted electrons / J metal B Threshold frequency f0 Frequency f / Hz EK = hf - Φ Work function, Φ Gradient of each graph = Planck’s constant, h. John Parkinson

Work Function and Photoelectric Emission
When u-v light is incident upon a zinc surface, each photon gives its energy to a single electron on the zinc surface. u-v photons have a high frequency. As a result they give enough energy to the electron to escape from the surface. The minimum energy needed to just remove an electron from a metal surface is called the work function, .

Low intensity u-v light will still cause electrons to be emitted
Low intensity u-v light will still cause electrons to be emitted. Because there are less photons per second there will be less electrons emitted per second. If the incident light has a lower frequency, each photon has less energy and so no electrons are emitted, irrespective of the intensity.

Einstein’s Photoelectric Equation
If the photon energy (E=hf) is greater than the work function (), any remaining energy becomes kinetic energy of the electron (= ½mv2). Thus Einstein stated... This is one version of Einstein's photoelectric equation. If hf is less than  nothing can happen hf =  + ½ mv2 m = mass of an electron v = speed of fastest electrons (ms-1)

Einstein’s theory was confirmed by Robert Millikan in 1916
Einstein’s theory was confirmed by Robert Millikan in He realised that if the clean metal emitting surface was given a positive potential, the electron emission could be stopped. Thus, if the p.d. applied was known, the KE ‘removed’ from the fastest electron can be found: We know... V = W / q  W = eV So... Link - PhET simulation V = stopping voltage e = charge on an electron = 1.6 x 10-19C  = Work function (Joules) hf =  + eV

Testing Einstein's Photoelectric Equation
Einstein’s photoelectric equation can be rearranged to give... Thus plotting a graph of V against f enables us to determine Plank’s constant. V = h f -  e e

Incident radiation is shone onto a photoelectric cell with a low work function
This causes electrons to be emitted from the larger emitting electrode. If they reach the small receiving electrode a current is detected on the ammeter The stopping voltage V is increased until zero current flows through the ammeter. The p.d. has made it impossible for even the fastest electrons to escape from the large electrode. The experiment is repeated with different frequency incident radiation and a set of values for frequency and stopping voltage are collected.

Results:

Q. Explain why this graph will always have the same gradient, whatever metal is used for the emitting electrode. Q. Explain how you would determine a value for the work function of the metal used to produce the graph above. Q. Wave theory suggests that Supported or Contradicted by quantum theory? Any frequency can emit electrons from a certain metal surface Current depends on intensity (i.e. the number of photons per unit time) Maximum energy of electrons is independent of frequency Maximum energy of electrons would depend on intensity

Photocurrents If the metal surface and frequency of incident radiation are both kept constant, a graph can be plotted showing how the photoelectric current (photocurrent) in a photocell varies with applied p.d. (voltage). Consider these situations and explain the photocurrent that will flow in each case: + - - +

Photocurrent Applied p.d. 0 + Saturation current
Saturation current Stopping potential, Vs

Photocurrent Applied p.d. 0 +
High intensity Low intensity We find the stopping voltage stays the same no matter what the intensity of the light source. The increase in current is due to more electrons being emitted. The intensity of light has no effect on the maximum energy.

Photocurrent Applied p.d. 0 + Low frequency (red)
Low frequency (red) Higher frequency (blue)

The electron-volt (revision)
Quantum Phenomena Breithaupt pages 30 to 43 The electron-volt (revision) The electron-volt (eV) is equal to the kinetic energy gained by an electron when it is accelerated by a potential difference of one volt. 1 eV = 1.6 x J Question: Calculate the energy in electron-volts of a photon of ultraviolet light of frequency 8 x 1014 Hz. (h = 6.63 x Js) April 8th, 2010

Ionisation An ion is a charged atom
Quantum Phenomena Breithaupt pages 30 to 43 Ionisation An ion is a charged atom Ions are created by adding or removing electrons from atoms The diagram shows the creation of a positive ion from the collision of an incoming electron. Ionisation can also be caused by: nuclear radiation – alpha, beta, gamma heating passing an electric current through a gas (as in a fluorescent tube) incoming electron April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Ionisation energy Ionisation energy is the energy required to remove one electron from an atom. Ionisation energy is often expressed in eV. The above defines the FIRST ionisation energy – there are also 2nd, 3rd etc ionisation energies. April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Excitation Excitation is the promotion of electrons from lower to higher energy levels within an atom. In the diagram some of the incoming electron’s kinetic energy has been used to move the electron to a higher energy level. The electron is now said to be in an excited state. Atoms have multiple excitation states and energies. incoming electron April 8th, 2010

Question An electron with 6 x 10-19J of kinetic energy can cause
Quantum Phenomena Breithaupt pages 30 to 43 Question An electron with 6 x 10-19J of kinetic energy can cause (a) ionisation or (b) excitation in an atom. If after each event the electron is left with (a) 4 x 10-19J and (b) 5 x 10-19J kinetic energy calculate in eV the ionisation and excitation energy of the atom. April 8th, 2010

Electron energy levels in atoms
Quantum Phenomena Breithaupt pages 30 to 43 Electron energy levels in atoms Electrons are bound to the nucleus of an atom by electromagnetic attraction. A particular electron will occupy the nearest possible position to the nucleus. This energy level or shell is called the ground state. It is also the lowest possible energy level for that electron. Only two electrons can exist in the lowest possible energy level at the same time. Further electrons have to occupy higher energy levels. April 8th, 2010

Electron energy levels in atoms
Quantum Phenomena Breithaupt pages 30 to 43 Electron energy levels in atoms Energy levels are measured with respect to the ionisation energy level, which is assigned 0 eV. All other energy levels are therefore negative. The ground state in the diagram opposite is eV. Energy levels above the ground state but below the ionisation level are called excited states. Different types of atom have different energy levels. April 8th, 2010

emitted photon energy = hf = E1 – E2
Quantum Phenomena Breithaupt pages 30 to 43 De-excitation Excited states are usually very unstable. Within about s the electron will fall back to a lower energy level. With each fall in energy level (level E1 down to level E2) a photon of electromagnetic radiation is emitted. emitted photon energy = hf = E1 – E2 April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Energy level question Calculate the frequencies of the photons emitted when an electron falls to the ground state (at 10.4 eV) from excited states (a) 5.4 eV and (b) 1.8 eV. (h = 6.63 x Js) April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Complete: transition photon E1 / eV E2 / eV f / PHz λ / nm 5.4 10.4 1.20 250 1.8 2.08 144 0.871 344 4.5 0.653 459 April 8th, 2010

Excitation using photons
Quantum Phenomena Breithaupt pages 30 to 43 Excitation using photons An incoming photon may not have enough energy to cause photoelectric emission but it may have enough to cause excitation. However, excitation will only occur if the photon’s energy is exactly equal to the difference in energy of the initial and final energy level. If this is the case the photon will cease to exist once its energy is absorbed. April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Fluorescence The diagram shows an incoming photon of ultraviolet light of energy 5.7eV causing excitation. This excited electron then de-excites in two steps producing two photons. The first has energy 0.8eV and will be of visible light. The second of energy 4.9eV is of invisible ultraviolet of slightly lower energy and frequency than the original excitating photon. This overall process explains why certain substances fluoresce with visible light when they absorb ultraviolet radiation. Applications include the fluorescent chemicals are added as ‘whiteners’ to toothpaste and washing powder. Electrons can fall back to their ground states in steps. April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Fluorescent tubes A fluorescent tube consists of a glass tube filled with low pressure mercury vapour and an inner coating of a fluorescent chemical. Ionisation and excitation of the mercury atoms occurs as the collide with each other and with electrons in the tube. The mercury atoms emit ultraviolet photons. The ultraviolet photons are absorbed by the atoms of the fluorescent coating, causing excitation of the atoms. The coating atoms de-excite and emit visible photons. See pages 37 and 38 for further details of the operation of a fluorescent tube April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Line spectra A line spectrum is produced from the excitation of a low pressure gas. The frequencies of the lines of the spectrum are characteristic of the element in gaseous form. Such spectra can be used to identify elements. Each spectral line corresponds to a particular energy level transition. April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Question Calculate the energy level transitions (in eV) responsible for (a) a yellow line of frequency 5.0 x 1014 Hz and (b) a blue line of wavelength 480 nm. April 8th, 2010

The hydrogen atom - 21.8 Quantum Phenomena Breithaupt pages 30 to 43
April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 - 21.8 With only one electron, hydrogen has the simplest set of energy levels and corresponding line spectrum. Transitions down to the lowest state, n=1 in the diagram, give rise to a series of ultraviolet lines called the Lyman Series. Transitions down to the n=2 state give rise to a series of visible light lines called the Balmer Series. Transitions down to the n=3, n=4 etc states give rise to sets of infra-red spectral lines. April 8th, 2010

The discovery of helium
Quantum Phenomena Breithaupt pages 30 to 43 The discovery of helium Helium was discovered in the Sun before it was discovered on Earth. Its name comes from the Greek word for the Sun ‘helios’. A pattern of lines was observed in the Sun’s spectrum that did not correspond to any known element of the time. In the Sun helium has been produced as the result of the nuclear fusion of hydrogen. Subsequently helium was discovered on Earth where it has been produced as the result of alpha particle emission from radioactive elements such as uranium. April 8th, 2010

The wave like nature of light
Quantum Phenomena Breithaupt pages 30 to 43 The wave like nature of light Light undergoes diffraction (shown opposite) and displays other wave properties such as polarisation and interference. By the late 19th century most scientists considered light and other electromagnetic radiations to be like water waves April 8th, 2010

The particle like nature of light
Quantum Phenomena Breithaupt pages 30 to 43 The particle like nature of light Light also produces photoelectric emission which can only be explained by treating light as a stream of particles. These ‘particles’ with wave properties are called photons. April 8th, 2010

The dual nature of electromagnetic radiation
Quantum Phenomena Breithaupt pages 30 to 43 The dual nature of electromagnetic radiation Light and other forms of electromagnetic radiation behave like waves and particles. On most occasions one set of properties is the most significant The longer the wavelength of the electromagnetic wave the more significant are the wave properties. Radio waves, the longest wavelength, is the most wavelike. Gamma radiation is the most particle like Light, of intermediate wavelength, is best considered to be equally significant in both April 8th, 2010

Matter waves λ = h / p λ = h / mv
Quantum Phenomena Breithaupt pages 30 to 43 Matter waves In 1923 de Broglie proposed that particles such as electrons, protons and atoms also displayed wave like properties. The de Broglie wavelength of such a particle depended on its momentum, p according to the de Broglie relation: λ = h / p As momentum = mass x velocity = mv λ = h / mv This shows that the wavelength of a particle can be altered by changing its velocity. April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Question 1 Calculate the de Broglie wavelength of an electron moving at 10% of the speed of light. [me = 9.1 x kg] (h = 6.63 x Js; c = 3.0 x 108 ms-1) April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Question 2 Calculate the de Broglie wavelength of a person of mass 70 kg moving at 2 ms-1. (h = 6.63 x Js) April 8th, 2010

Quantum Phenomena Breithaupt pages 30 to 43 Question 3 Calculate the effective mass of a photon of red light of wavelength 700 nm. (h = 6.63 x Js) April 8th, 2010

Evidence for de Broglie’s hypothesis
Quantum Phenomena Breithaupt pages 30 to 43 Evidence for de Broglie’s hypothesis A narrow beam of electrons in a vacuum tube is directed at a thin metal foil. On the far side of the foil a circular diffraction pattern is formed on a fluorescent screen. A pattern that is similar to that formed by X-rays with the same metal foil. Electrons forming a diffraction pattern like that formed by X-rays shows that electrons have wave properties. The radii of the circles can be decreased by increasing the speed of the electrons. This is achieved by increasing the potential difference of the tube. April 8th, 2010

Energy levels and electron waves
Quantum Phenomena Breithaupt pages 30 to 43 Energy levels and electron waves An electron in an atom has a fixed amount of energy that depends on the shell it occupies. Its de Broglie wavelength has to fit the shape and size of the shell. April 8th, 2010

Potassium Magnesium Aluminium Max Ek / eV 2 1 0 5 10 15 f / Hz 1014 ©
Max Ek / eV 1 2 Potassium Magnesium Aluminium John Parkinson

Summary For incident radiation of a given frequency, the number of electrons emitted per second is proportional to the intensity of the radiation. For any metal there is a minimum threshold frequency, f0, of the incident radiation, below which no emission of electrons takes place, no matter what the intensity of the incident radiation is or for how long it falls on the surface. Electrons emerge with a range of velocities from zero up to a maximum. The maximum kinetic energy, Ek, is found to depend linearly on the frequency of the radiation and to be independent of its intensity. Electron emission takes place immediately after the light shines on the metal with no detectable time delay . John Parkinson

Einstein’s Theory hf =  + ½ mv2 hf equals the energy of each photon
The photoelectric effect is interpreted with photons and the conservation of energy with the equation: hf =  + ½ mv2 hf equals the energy of each photon Source:

Kinetic energy of emitted electron vs. Light frequency
Higher-frequency photons have more energy, so they should make the electrons come flying out faster; thus, switching to light with the same intensity but a higher frequency should increase the maximum kinetic energy of the emitted electrons. If you leave the frequency the same but crank up the intensity, more electrons should come out (because there are more photons to hit them), but they won't come out any faster, because each individual photon still has the same energy. And if the frequency is low enough, then none of the photons will have enough energy to knock an electron out of an atom. So if you use really low-frequency light, you shouldn't get any electrons, no matter how high the intensity is. Whereas if you use a high frequency, you should still knock out some electrons even if the intensity is very low. Source: content/PhyAPB/lessonnotes/dualnature/ photoelectric.asp

Simple Photoelectric Experiment
Source:

Photoelectric Effect Applications

Applications The Photoelectric effect has numerous applications, for example night vision devices take advantage of the effect. Photons entering the device strike a plate which causes electrons to be emitted, these pass through a disk consisting of millions of channels, the current through these are amplified and directed towards a fluorescent screen which glows when electrons hit it. Image converters, image intensifiers, television camera tubes, and image storage tubes also take advantage of the point-by-point emission of the photocathode. In these devices an optical image incident on a semitransparent photocathode is used to transform the light image into an “electron image.” The electrons released by each element of the photoemitter are focused by an electron-optical device onto a fluorescent screen, reconverting it in the process again into an optical image

Applications: Night Vision Device

Photoelectric Effect Applications
Photoelectric Detectors In one type of photoelectric device, smoke can block a light beam. In this case, the reduction in light reaching a photocell sets off the alarm. In the most common type of photoelectric unit, however, light is scattered by smoke particles onto a photocell, initiating an alarm. In this type of detector there is a T-shaped chamber with a light-emitting diode (LED) that shoots a beam of light across the horizontal bar of the T. A photocell, positioned at the bottom of the vertical base of the T, generates a current when it is exposed to light. Under smoke-free conditions, the light beam crosses the top of the T in an uninterrupted straight line, not striking the photocell positioned at a right angle below the beam. When smoke is present, the light is scattered by smoke particles, and some of the light is directed down the vertical part of the T to strike the photocell. When sufficient light hits the cell, the current triggers the alarm. Source:

Photoelectric Smoke Detector
Source:

Applications Solar panels are nothing more than a series of metallic plates that face the Sun and exploit the photoelectric effect. The light from the Sun will liberate electrons, which can be used to heat your home, run your lights, or, in sufficient enough quantities, power everything in your home. Source: picsolarpannelsmatt.jpg

The Wave Nature of Matter
We have seen that light behaves both like a wave (it diffracts) and like a particle (in explaining photoelectric emission). It has wave particle duality. Demo 1: Shine a beam of laser light (a wave) through a single diffraction grating (like a very fine gauze) then through many gratings crossed at angles to each other. Demo 2: Fire a beam of electrons (particles of matter) at a thin piece of graphite using a cathode ray tube. Wave - particle duality is the ability of something to exhibit both wave and particle behaviour.

VA Cathode Control grid Anode Graphite foil Flourescent screen

De Broglie’s Equation Considering a photon of light, in 1924 Prince Louis Victor de Broglie equated Einstein’s mass-energy relation and Planck’s equation: E = mc2 and E = hf thus... mc2 = hf so mc2 = h c λ   so or But c = f λ λ = h mc λ = h p

This is the de Broglie equation.
( Where h = Planck’s constant = 6.6 x 10-34Js ) By analogy, this equation can be applied to any other particle of matter. Thus de Broglie’s Hypothesis: Q1 A year 13 student runs with joy to his physics lesson. If he runs at 5 ms-1 and has mass 60kg, determine the de Broglie wavelength of his motion. Comment upon your answer. All matter can behave like a wave, with the wavelength given by Plank’s constant divided by the matter’s momentum.

Q2 An electron is accelerated in a cathode ray tube through a potential difference of 2kv.
Determine the velocity of the electron (me = 9.11 x kg) 2.65 x 107 ms-1 Determine the de Broglie wavelength of the electron. 2.7 x m

Energy Levels E.g. h = Planck constant = 6.6 x m2kgs-1 E = hf hf = E2 - E1

Experiment: Observing emission spectra
Emission Line Spectra Atoms of a gas emit e-m radiation if they become excited. This means the electrons jump to a higher energy level and then fall back, losing potential energy and emitting it as a photon of e-m radiation. This will have a frequency according to ΔE = hf Experiment: Observing emission spectra Place a slit in front of a hydrogen lamp. View the light through a diffraction grating or by refracting it through a prism. A neon discharge lamp is a tube containing neon atoms which are being bombarded by high speed electrons (accelerated by a high p.d.).

Results: Conclusion Explain why... only certain colours are seen for any particular lamp. coloured fringes are produced This is the image seen when light from one lamp is diffracted through a grating:

Neon

Mercury

Argon

Xenon Images from this website

Hydrogen... Helium... Neon...

Absorption spectrum If light with a continuous spectrum of frequencies (a filament light bulb approximately emits this) is shone into a gas its photons will interact with electrons in the gas atoms, boosting them to a higher energy level. Thus the gas will absorb only certain frequencies of the light. These frequencies will be missing from the light that passes through.

Absorption spectrum for Hydrogen.
Q. Why des the intensity not fall to zero for the absorption lines? Energy is re-emitted as photons in all directions

Fraunhoffer Lines This effect is visible in the Fraunhoffer lines seen in spectra of light from the Sun. The Sun emits a virtually continuous spectrum. The absorption lines are due to gases in the Sun’s atmosphere absorbing some frequencies of photon. Hence astronomers can deduce what gases are in a star’s atmosphere as well as their proportions.

The ‘Electron in a Box’ Model of an Atom
We have seen how the Bohr model of an atom suggests that the atom has discreet energy levels. Thus the energy of the electrons and of the atom is not continuous. Next we will look at how this idea is compatible with the de Broglie hypothesis. According to de Broglie, the electron must have wave properties. So how does this wave fit into an atom...?

We can create a quantum model of an electron trapped in an atom by considering the electron as a standing wave travelling back and forth across a box of side L: Clearly, if there are nodes at either end, the electron can only have certain fixed wavelengths and frequencies. i.e. λ = 2L (fundamental) λ = L (2nd harmonic) λ = 2/3 L (3rd harmonic) So in general λ = 2L n Java applet

According to de Broglie...
The KE of the electron is given by KE = ½ mv2 Substituting the equations for v and λ gives… The energy of the electron ‘wave’ is clearly quantised (it has discreet values depending upon n), thus fitting scientific observations e.g. work function in the photoelectric effect. λ = h = h  v = h p mv mλ n = integer h, m and L are all constants. KE = n2 h2 8mL2

Limitations of the Bohr Model of the Atom
The Bohr model of the atom worked well with basic ideas of quantum theory (as we have seen). However it has limitations. E.g. It only works well with the Hydrogen atom It does not predict intensities of different spectral lines.

Schrödinger's Model of the Atom
Schrödinger developed a new model that accounted for all these limitations. He suggested that... electrons exist in the atom with a position determined by the wavefunction (ψ – psi), a function of position and time. the position of the electron at any time is undefined the probability of the finding the electron at any particular position in the atom is determined by the square of the amplitude of the wavefunction i.e. ψ2 Thus for a given energy there are some places where the electron is more likely to exist. This can be represented by ‘probability clouds’.

Additional note: This illustrates an important feature of quantum mechanics. Its outcomes are probabilities and not certainties; it is a probabilistic model. Classical mechanics (e.g. Newtons laws) are deterministic: the future is determined by the past and is therefore predictable with some certainty!

Heisenberg’s Uncertainty Principle
Consider one electron amongst a beam of electrons moving with identical momentum towards a narrow slit: If the slit is wide, the electron will pass straight through without diffracting. Hence we can know its direction and thus its momentum beyond the slit. However we cannot be sure where exactly it passed through the slit so the uncertainty in position is large.

If the slit width is similar to the diameter of the electron, it will diffract as it passes through.
So we are not sure about its direction and thus momentum although we can be quite sure of its position. This is an example of the Heisenberg uncertainty principle which states that... Note: This is a fundamental property of the universe and is nothing to do with our ability to measure accurately. It is not possible to measure the exact position and the exact momentum of a particle at the same time.

This can also be written in terms of energy and time:
This imposes a minimum uncertainty on the product of the uncertainties of position and momentum: This can also be written in terms of energy and time: Clearly the value of h is so small that the effects of this uncertainty are not seen in everyday events. Δp Δx ≥ h h = Plank’s constant ΔE Δt ≥ h

E.g. Quantum tunnelling As a result of the uncertainty principle energy can be ‘borrowed’ to overcome potential barriers: Alpha particles should not be able to escape from the nucleus due to the ‘strong force’ between quarks. However they escape by borrowing energy and paying it back within a time limit defined by the equation. Two protons (hydrogen nuclei) should not be able to fuse together in the Sun at its current density and temperature. They do fuse together by borrowing energy and then paying it back later on. You tube link

Q1 Consider a football moving through the air. If the uncertainty in its momentum is 2.25 x 10-2 kgms-1, Determine the uncertainty in its position (Δx). Δx ≥ +/- 2.33x10-33m Δx Δp ≥ h so… Δx ≥ h 4π Δp Δx ≥ x 10-34 4π x (2.25 x 10-2)

Q2 A proton in the LHC takes 1.0 x 10-10s to collide. Assuming a 1% uncertainty in measurement of time, determine the uncertainly in measurements of proton energy (ΔE). 1% uncertainty in time = 1/100 x (1.0 x 10-10) = +/- 1.0 x s ΔE Δt ≥ h ΔE ≥ x 10-34 4π x (1.0 x 10-12) so… ΔE ≥ h 4π Δt ΔE ≥ +/- 5.3 x J

Q3 Assuming 1% uncertainty in velocity, determine the uncertainty in the position of a proton moving at 2.5 x 107ms-1 Close to the speed of light (3.0 x 108 ms-1) ( mp = x kg )

Q4 Outline by reference to position and momentum how the Schrödinger model of the hydrogen atom is consistent with the Heisenberg uncertainty principle. - the Schrödinger model assigns a wave function to the electron that is a measure of the probability of finding it somewhere; - therefore the position of the electron is uncertain; - resulting in an uncertainty in its momentum;

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Probability Waves From the de Broglie hypothesis we can see that our picture of the electron as a small ‘ball’ is too simplistic. It

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