Section 2 - Electromagnetic Radiation and Quantum Phenomena - complete

-if you shine light of a high enough frequency onto the surface of a metal, the metal will emit electrons
- for most metals, this frequency falls in the UV range
1) free electrons on the surface of the metal absorb energy from the light
2) if an electron absorbs enough energy, the bonds holding it to the metal break and the electron is released
3) this is called the photoelectric effect and the free electrons are called photoelectrons

3 Main Conclusions from this ^
1 - for a given metal, no photoelectrons are emitted if the radiation has a frequency below a certain value (threshold frequency)
2 - the photoelectrons are emitted with a variety of kinetic energies randing from 0 to some maximum value. the value of maximum kinetic energy increases with the frequency of the radiation, and is unaffected by the intensity of the radiation
3 - the number of photoelectrons emitted per second is proportional to the intensity of the radiation

Wave Theory:
- for a particular frequency of light, the energy carried is proportional to the intensity of the beam
- the energy carried by the light would be spread evenly over the wavefront
- each free electron on the surface of the metal would gain a bit of energy from each incoming wave
- gradually, each electron would gain enough energy to leave the metal
SO
- the higher the intensity of the wave, the more energy it should transfer to each electron (the kinetic energy should increase with intensity) - there is no explanation for the kinetic energy depending only on the frequency
- there is also no explanation for the threshold frequency - according to wave theory, the electrons should be emitted eventually, no matter what the frequency is

- einstein suggested that EM waves (and the energy they carry) exist in discrete packets called photons
- the energy carried by one of these photons is: E = hf = hc/wavelength
(where h = 6.63 x10^-34 Js and c = 3x10⁸ ms^-1)
- einstein saw these photons of light as having a one-on-one, particle-like interaction with an electron in a metal surface
- a photon would transfer all its energy to one, specific electron

According to the photon model:
- when light hits its surface, the metal is bombarded by photons
- if one of these photons collides with a free electron, the electron will gain energy equal to hf

- before an electron can leave the surface of the metal, it needs enough energy to break the bonds holding it there
- this energy is called the work function (symbol phi) and its value depends on the metal

TF
- if the energy gained by an electron from a photon is greater than the work function, the electron is emitted
- if it isnt, the metal will heat up but no electrons will be emitted
- since, for electrons to be released, hf > phi, the threshold frequency must be f = phi / h 

MKE
- the energy transferred to an electron is hf
- the kinetic energy the electron will be carrying when it leaves the metal is hf minus and energy its lost on the way out
- electrons deeper down in the metal lose more energy that the electrons on the surface, which explains the range of energies
- the minimum amount of energy it can lose is the work function, so the MKE of a photon (E_k(max)) is given by the photoelectric equation
hf = phi + MKE where MKE = 1/2mv_max² 
- the KE of the electrons is independent of the intensity (the no. of photons per second on an area), as they can only absorb one photon at a time
- increasing the intensity just means more photons per second

- the maximum kinetic energy can be measured using the idea of stopping potential
- the emitted electrons are made to lose their energy by doing work against an applied potential difference
- the stopping potential (V_s) is the p.d. needed to stop the fastest moving electrons, with E_{k (max)}
- the work done by the p.d. in stopping the fastest electrons is equal to the energy they were carrying:
eV_s = E_{k (max)}
- where e = charge on the electron = 1.60 x 10^-19 C
- where V_s = stopping potential
- where E_{k (max)} is measured in J (joules)

- electrons in an atom can only exist in certain well-defined energy levels
- each level is given a number, with n = 1 representing the ground state
- electrons can move down energy levels by emitting a photon
- since these transitions are between definite energy levels, the energy of each photon can only take a certain allowed value
- the energies involved are so tiny that it makes sense to use a more appropriate unit than the joule
- the electronvolt (eV) is defined as: "the kinetic energy carried by an electron after it has been accelerated through a potential difference of 1 volt"
- energy gained by electron (eV) = accelerating voltage (V)
- the energy carried by each photon is equal to the difference in energies between the two levels
- the equation (change in E = E₂ - E₁ = hf = hc/wavelength) shows a transition between a higher energy level (n = 2) where the electrons have energy E₂ and a lower energy level (n=1) with electrons of energy E₁
- electrons can also move up energy levels if they absorb a photon with the exact energy difference between the two levels
- the movement of an electron to a higher energy level is called excitation
- if an electron is removed from an atom, we say the atom is ionised
- the energy of each energy level within an atom gives the amount of energy needed to remove an electron in the level from the atom
- the ionisation energy of an atom is the amount of energy needed to completely remove an electron from the atom from the ground state (n = 1)

- fluorescent tubes contain mercury vapour, across which an initial high voltage is applied
- this high voltage accelerated fast-moving free electrons that ionise some of the mercury atoms, producing more free electrons
- when this flow of free electrons collides with electrons in other mercury atoms, the electrons in the mercury aroms are excited to higher energy levels
- when these excited electrons return to their ground states, they emit photons in the UV range
- a phosphor coating on the inside of the tube absorbs these photons, exciting its electrons to much higher orbits
- these electrons then cascade down the energy levels, emitting many lower energy photons in the form of visible light

- if you split the light from a fluorescent tube with a prism or a diffraction grating, you get a line spectrum
- a line spectrum is seen as a series of bright lines against a black background
- each line corresponds to a particular wavelength of light emitted by the source
- since only certain photon energies are allowed, you only see the wavelengths corresponding to these energies

- the spectrum of white light is continuous
- if you split the light up with a prism, the colours all merge into each other (there aren't any gaps in the spectrum)
- hot things emit a continuous spectrum in the visible and infrared
- all the wavelengths are allowed because the electrons are not confined to energy levels in the object producing the continuous spectrum
- the electrons are not bound to atoms and are free

- you get a line absorption spectrum when light with a continuous spectrum of energy (white light) passes through a cool gas
- at low temperatures, most of the electrons in the gas atoms will be in their ground states
- the electrons can only absorb photons with energies equal to the difference between two energy levels
- photons of the corresponding wavelengths are absorbed by the electrons to excited them to higher energy levels
- these wavelengths are then missing from the continuous spectrum when it comes out the other side of the gas
- you see a continuous spectrum with black lines in it corresponding to the absorbed wavelengths
- if you compare the absorption and emission spectra of a particular gas, the black lines in the absorption spectrum match up to the bright lines in the emission spectrum

- interference and diffraction show light as a wave
- light produces interference and diffraction patterns, alternating bands of dark and light
- these can only be explained using waves interfering constructively (when two waves overlap in phase) or interfering destructively (when two waves are out of phase)

- the photoelectric effect shows light behaving as a particle
- einstein explained the results of photoelectricity experiments by thinking of the beam of light as a series of particle-like photons
- if a photon of light is a discrete bundle of enegry, then it can interact with an electron in a one-to-one way
- all the energy in the photon is given to one electron

- Louis de Broglie made this statement in this PhD thesis:
"if 'wave-like' light showed particle properties (photons), 'particles' like electrons should be expected to show wave-like properties"
- the de Broglie equation relates a wave property (wavelength) to a moving particle property (momentum, mv). h = planck's constant = 6.63x10^-34 Js
- wavelength = planck's constant / momentum
- the de Broglie wave of a particle can be interpreted as a 'probability wave' (the probability of finding a particle at a point is directly proportional to the square of the amplitude of the wave at that point)
- experiments have confirmed the wave nature of electrons

- diffraction patterns are observed when accelerated electrons in a vacuum tube interact with the spaces in a graphite crystal
- this confirms that electrons show wave-like properties
- according to wave theory, the spread of the lines in the diffraction pattern increases if the wavelength of the wave is greater
- in electron diffraction experiments, a smaller accelerating voltage, i.e. slower electrons, gives more widely-spaced rings
- increase the electron speed (and therefore the electron momentum) and the diffraction pattern circles squash together towards the middle
- this fits in with the de Broglie equation above - if the momentum is greater, the wavelength is shorter and the spread of the lines is smaller
- in general, wavelength for electrons accelerated in a vacuum tube is about the same size as electromagnetic waves in the x-ray part of the spectrum
- if particles with a greater mass (e.g. neutrons) were travelling at the same speed as the electrons, they would show a more tightly-packed diffraction pattern
- this is because a neutron's mass (and therefore its momentum) is much greater than an electron's, and so a neutron has a shorter de Broglie wavelength

- you only get diffraction if a particle interacts with an object of about the same size as its de Broglie wavelength
- a tennis ball, for example, with a mass of 0.058kg and speed of 100ms^-1 has a de Broglie wavelength of 10^-34m
- that ^ is 10¹⁹ times smaller than the nucleus of an atom meaning there is nothing small enough for it to interact with

EXAMPLE
An electron of mass 9.11 x 10^-31 kg is fired from an electron gun at 7 x 10⁶ ms^-1 (3 s.f.). What size object will the electron need to interact with in order to diffract?

wavelength = planck's constant / (mass x speed)
= 6.63 x 10^-34 / (9.11 x 10^-31 x 7 x 10⁶)
= 6.63 x 10^-34 / 63.77 x 10^-25)
= 1.04 x 10^-10

- de Broglie first hypothesised wave-particle duality to explain observations of light acting as both a particle and a wave
- this theory wasn't accpeted straight away
- other scientists had to evaluate de Broglie's theory through peer review before he published it and then it was tested with experiments
- once enough evidence was found to back it up, the theory was accepted and validated by the scientific community
- scientists' understanding of the nature of matter has changed over time through this process of hypothesis and validation
- de broglie's theory is accepted to be true until any new conflicting evidence comes along