22.1 Energy and momentum of a photon
Definitions
Photon a quantum (discrete packet) of energy of electromagnetic radiation.
Electronvolt (eV) the energy gained by an electron when it is accelerated through a potential difference of 1 V (1 eV = 1.60 × 10-19 J).
Planck constant (h) the constant relating photon energy to frequency: h = E/f = 6.63 × 10-34 J s. Key points Photon energy is proportional to frequency and inversely proportional to wavelength . Intensity of a beam = number of photons per second per unit area × energy of each photon. A photon has no mass but carries momentum p = E/c; absorbing or reflecting light exerts a force. eV to J: multiply by e (1.60 × 10-19 ); J to eV: divide by e. An electron accelerated through V gains kinetic energy eV . Visible photons have energies of roughly 1.8 to 3.1 eV (red to violet).
E f
gradient = h
E λ
E ∝ 1/λ
Photon energy against frequency is a straight line through the origin (gradient h); against wavelength it is an inverse curve.
Watch out! Short wavelength means high photon energy (UV, X-rays), not low. Convert eV to J before using E = hf with h in J s.
22.2 Photoelectric effect
Definitions
Photoelectric effect the emission of electrons from the surface of a metal when electromagnetic radiation of high enough frequency falls on it.
Work function (Φ) the minimum energy needed to remove an electron from the surface of a metal.
Threshold frequency (f0 ) the minimum frequency of electromagnetic radiation that will release electrons from the surface of a metal.
Threshold wavelength (λ0 ) the maximum wavelength of electromagnetic radiation that will release electrons from the surface of a metal. Key points Each electron absorbs one photon : all of the photon energy or none (one-to-one interaction). Below f0 no electrons are emitted, however intense the light; above f0 emission is instantaneous . Max KE depends on frequency only, not intensity: photon energy minus work function. Electrons deeper in the metal lose extra energy, so emitted KE ranges from 0 up to KEmax . Photoelectric current ∝ intensity (more photons per second, more electrons per second), for f > f0 . These features show light is a particle (photon); a wave model predicts emission at any frequency after a delay. Stopping potential Vs : the p.d. that just stops the fastest electrons, so eVs = KEmax . Equations hf = Φ + ½mvmax 2
Einstein's photoelectric equation
J
Φ = hf0 = hc / λ0
work function and threshold
J
eVs = ½mvmax 2
stopping potential measures KEmax
J
KE max
f
f₀
-Φ
gradient = h
KEmax against f: straight line, gradient h, x-intercept f0 , extrapolated y-intercept -Φ. Every metal gives a parallel line.
cathode anode
light, f
e⁻
A
V
Photocell: light on the cathode releases electrons; the reverse p.d. that makes the current just zero is the stopping potential.
Watch out! Greater intensity gives more photoelectrons (bigger current), never faster ones. The equation uses the maximum KE; most emitted electrons have less. Threshold wavelength is a maximum, threshold frequency a minimum.
22.3 Wave-particle duality
Definitions
Wave-particle duality the idea that light and matter show both wave properties and particle properties.
De Broglie wavelength the wavelength associated with a moving particle: λ = h/p, where p is its momentum. Key points Light as a particle : the photoelectric effect. Light as a wave : diffraction and interference. Electron diffraction is evidence that particles (electrons) behave as waves.Electrons through thin graphite : layer spacing (~10-10 m) is similar to λ, giving diffraction rings on the screen. Higher accelerating p.d. gives faster electrons, shorter λ , so smaller rings . Electron accelerated through V: ½mv2 = eV, so p = √(2meV). Diffraction is only noticeable when λ is comparable with the gap or spacing.
gun
graphite
screen
rings
Electron diffraction tube: electrons from the gun pass through graphite and form concentric rings on the fluorescent screen.
λ p
λ = h/p
faster: smaller λ
De Broglie wavelength against momentum: an inverse relationship (λp = h).
Watch out! Faster electrons have shorter wavelengths and diffract less : the rings shrink. Use p = mv for the particle, not E/c (that is for photons only).
22.4 Energy levels in atoms and line spectra
Definitions
Energy level one of the discrete (quantised) energies that an electron in an isolated atom can have.
Ground state the lowest energy level of an atom.
Excitation the promotion of an electron to a higher energy level by absorbing a photon of exactly the right energy (or by collision).
Ionisation energy the minimum energy needed to remove an electron from the ground state of the atom completely (to the zero level).
Line spectrum a spectrum consisting of discrete wavelengths only, each corresponding to a transition between two energy levels. Key points Energy levels are negative ; zero corresponds to a free electron at infinity. Levels crowd together near the top. Downward transition: photon emitted with energy exactly equal to the gap. Upward: photon of exactly that energy absorbed. Discrete levels give discrete photon energies , so discrete wavelengths: line spectra, not continuous. Emission spectrum : bright lines on a dark background, from a hot low-pressure gas.Absorption spectrum : dark lines in a continuous spectrum, where white light passes through cooler gas.Absorption lines are at the same wavelengths as that element's emission lines; re-emission is in all directions. Biggest gap = highest energy photon = shortest wavelength line. Number of possible lines from level n down: n(n-1)/2.
0 (free)
E₃
E₂
E₁ ground
UV
IR
visible
Energy levels: a drop from E3 to E1 emits one photon with hf = E3 - E1 ; the longest arrow gives the shortest wavelength.
emission
absorption
wavelength λ
Emission lines (bright on dark) and absorption lines (dark on a continuous spectrum) at the same wavelengths for one element.
Watch out! Photon energy is the difference between two levels, not the value of one level. Subtracting negative levels: -1.5 eV - (-3.4 eV) = +1.9 eV . Mind the signs. Only photons of exactly a gap energy are absorbed (unless they ionise the atom).