1# Light and matter 2 3## Photoelectric effect 4 5 6 7### Planck's equation 8 9$$E=hf,\quad f={c \over \lambda}$$ 10$$\therefore E={hc \over \lambda}$$ 11 12where 13$E$ is energy of a quantum of light (J) 14$f$ is frequency of EM radiation 15$h$ is Planck's constant ($6.63\times 10^{-34}\operatorname{J s}=4.12\times 10^{-15} \operatorname{eV s}$) 16 17### Electron-volts 18 19$$ 1 \operatorname{eV} = 1.6\times 10^{-19} \operatorname{J}$$ 20 21*Amount of energy an electron gains when it moves through a potential difference of 1V* 22 23- equivalent unit is Joule seconds (e.g. $h$) 24 25### Photoelectric effect 26 27- some metals becomes positively charged when hit with EM radiation 28- this is due to e- being ejected from surface of metal 29- *photocurrent* - flow of e- due to photoelectric effect 30- causes increase in current in a circuit 31- $V_{\operatorname{supply}}$ does not affect photocurrent 32- if $V_{\operatorname{supply}} \gt 0$, e- are attracted to collector anode. 33- if $V_{\operatorname{supply}} \lt 0$, e- are attracted to illuminated cathode, and $I\rightarrow 0$ 34- not all electrons have the same velocity - depends on ionisation energy (shell) 35 36#### Wave / particle (quantum) models 37wave model: 38 39- cannot explain photoelectric effect 40- $f$ is irrelevant to photocurrent 41- predicts that there should be a delay between incidence of radiation and ejection of e- 42 43particle model: 44 45- explains photoelectric effect 46- rate of photoelectron release is proportional to intensity of incident light 47- shining light on a metal "bombards" it with photons 48- no time delay 49- one photon releases one electron 50 51#### Work function and threshold frequency 52 53- *threshold frequency* $f_0$ - minimum frequency for photoelectrons to be ejected 54- if $f \lt f_0$, no photoelectrons are detected 55 56- Einstein: energy required to eject photoelectron is constant for each metal 57- *work function* $\phi$ - minimum energy required to release photoelectrons 58- $\phi$ is determined by strength of bonding 59 60$$\phi=hf_0$$ 61 62#### $E_K$ of photoelectrons (stopping energy) 63 64$$E_{\operatorname{k-max}}=hf - \phi$$ 65 66where 67$E_k$ is max energy of an emmitted photoelectron 68$f$ is frequency of incident photon (**not** emitted electron) 69$\phi$ is work function ("latent" energy) 70 71Gradient of a frequency-energy graph is equal to $h$ 72y-intercept is equal to $\phi$ 73voltage $V$ in circuit is indicative of max kinetic energy in eV 74 75#### Stopping potential $V_0$ 76 77Smallest voltage to achieve minimum current 78 79$$V_0 = {E_{K \operatorname{max}} \over q_e} = {{hf - \phi} \over q_e}$$ 80 81## Wave-particle duality 82 83### Double slit experiment 84Particle model allows potential for photons to interact as they pass through slits. However, an interference pattern still appears when a dim light source is used so that only one photon can pass at a time. 85 86## De Broglie's theory 87 88$$\lambda = {h \over \rho} = {h \over mv}$$ 89 90- theorised that matter may display both wave- and particle-like properties like light 91- predict wavelength of a particle with $\lambda = {h \over \rho}$ where $\rho = mv$ 92- impossible to confirm de Broglie's theory of matter with double-slit experiment, since wavelengths are much smaller than for light, requiring an equally small slit ($< r_{\operatorname{proton}}$) 93- confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit) 94- also confirmed by Thomson - e- diffraction pattern resembles x-ray (wave) pattern 95- electron is only stable in orbit if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ 96- rearranging this, $2\pi r = n{h \over mv}$ (circumference) 97- therefore, stable orbits are those where circumference = whole number of e- wavelengths 98- if $2\pi r \ne n{h \over mv}$, interference occurs when pattern is looped and standing wave cannot be established 99 100 101 102### Photon momentum 103 104$$\rho = {hf \over c} = {h \over \lambda}$$ 105- if a massy particle (e.g. electron) has a wavelength, then anything with a wavelength must have momentum 106- therefore photons have (theoretical) momentum 107- to solve photon momentum, rearrange $\lambda = {h \over mv}$ 108 109## Spectral analysis 110 111 112### Absorption 113- Black lines in spectrum show light not reflected 114- Frequency of a photon emitted or absorbed can be calculated from energy difference: $E_2 – E_1 = hf$ or $= hc$ 115 116### Emission 117 118 119 120- Coloured lines show light being ejected from e- shells 121- Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$ 122- Bohr's model describes discrete energy levels 123- Energy is conserved (out = in) 124- Ionisation energy - minimum energy required to remove an electron 125- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$) 126 127## Light sources 128 129 130 131- **incandescent:** <10% efficient, broad spectrum 132- **LED:** semiconducting doped-Si diodes 133- - most electrons in *valence band* (energy level) 134- - provided energy, electrons can jump to *conduction band* and move through Si as current 135- - colour determined by $\Delta E$ between bands (shells), and type of doping 136- **laser:** gas atoms are excited 137- - *popular inversion* - most gas atoms are excited 138- - photons are released if stimulated by another photon of the right wavelength 139- **synchrotron:** - magnetically accelerates electrons 140- - extremely bright 141- - highly polarised 142- - emitted in short pulses 143- - broad spectrum 144 145## Quantum mechanics 146 147- uncertainty occurs in any measurement 148- inherent physical limit to absolute accuracy of measurements (result of wave-particle duality) 149- interaction between observer and object 150- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it 151 152### Indeterminancy principle 153 154$$\sigma E \sigma t \ge {h \over 4 \pi}$$ 155 156where $\sigma n$ is the uncertainty of $n$ 157 158**$\sigma E$ and $\sigma t$ are inversely proportional** 159 160Therefore, position and velocity cannot simultaneously be known with 100% certainty. 161 162### Single-slit diffraction 163 164- one photon passes through slit at any time (controlled by intensity) 165- diffraction pattern can be explained by wave front split into wavelets 166- diffraction can be represented as uncertainty of photonic momentum 167 168 169### Comparison with Bohr's model 170 171**Newtonian (deterministic) model** - current $x$ and $v$ are known, so future $x$ can be calculated 172 173**Quantum mechanical model** - electron clouds rather than discrete shells (electrons are not particlces). We can only calculate probability of an electron being observed at a particular position 174 175Newton's and Einsteins models work together 176 177### Photon-electron interaction 178 179When a photon collides with an electron, momentum is transferred to electron. 180 181$$\rho_{\text{photon}}={h \over \lambda}$$ 182$$E=\rho c$$ 183 184 185 186 187 188774 abc melbourne