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 50#### Work function and threshold frequency 51 52- *threshold frequency* $f_0$ - minimum frequency for photoelectrons to be ejected 53- if $f \lt f_0$, no photoelectrons are detected 54 55- Einstein: energy required to eject photoelectron is constant for each metal 56- *work function* $\phi$ - minimum energy required to release photoelectrons 57- $\phi$ is determined by strength of bonding 58 59$$\phi=hf_0$$ 60 61#### $E_K$ of photoelectrons (stopping energy) 62 63$$E_{\operatorname{k-max}}=hf - \phi$$ 64 65where 66$E_k$ is max energy of an emmitted photoelectron 67$f$ is frequency of incident photon (**not** emitted electron) 68$\phi$ is work function ("latent" energy) 69 70Gradient of a frequency-energy graph is equal to $h$ 71y-intercept is equal to $\phi$ 72 73#### Stopping potential $V_0$ 74$$V_0 = {E_{K \operatorname{max}} \over q_e} = {{hf - \phi} \over q_e}$$ 75 76## Wave-particle duality 77 78### Double slit experiment 79Particle 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. 80 81## De Broglie's theory 82 83$$\lambda = {h \over \rho} = {h \over mv}$$ 84 85- theorised that matter may display both wave- and particle-like properties like light 86- predict wavelength of a particle with $\lambda = {h \over \rho}$ where $\rho = mv$ 87- 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 ($< r_{\operatorname{proton}}$) 88- confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit) 89- also confirmed by Thomson - e- diffraction pattern resembles x-ray (wave) pattern 90- electron is only stable in orbit if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ 91- rearranging this, $2\pi r = n{h \over mv}$ (circumference) 92- therefore, stable orbits are those where circumference = whole number of e- wavelengths 93- if $2\pi r \ne n{h \over mv}$, interference occurs when pattern is looped and standing wave cannot be established 94 95 96 97### Photon momentum 98 99$$\rho = {hf \over c} = {h \over \lambda}$$ 100- if a massy particle (e.g. electron) has a wavelength, then anything with a wavelength must have momentum 101- therefore photons have (theoretical) momentum 102- to solve photon momentum, rearrange $\lambda = {h \over mv}$ 103 104## Spectral analysis 105 106 107### Absorption 108- Black lines in spectrum show light not reflected 109- Frequency of a photon emitted or absorbed can be calculated from energy difference: $E_2 – E_1 = hf$ or $= hc$ 110 111### Emission 112 113 114 115- Coloured lines show light being ejected from e- shells 116- Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$ 117- Bohr's model describes discrete energy levels 118- Energy is conserved (out = in) 119- Ionisation energy - minimum energy required to remove an electron 120- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$) 121 122## Light sources 123 124 125 126- **incandescent:** <10% efficient, broad spectrum 127- **LED:** semiconducting doped-Si diodes 128- - most electrons in *valence band* (energy level) 129- - provided energy, electrons can jump to *conduction band* and move through Si as current 130- - colour determined by $\Delta E$ between bands (shells), and type of doping 131- **laser:** gas atoms are excited 132- - *popular inversion* - most gas atoms are excited 133- - photons are released if stimulated by another photon of the right wavelength 134- **synchrotron:** - magnetically accelerates electrons 135- - extremely bright 136- - highly polarised 137- - emitted in short pulses 138- - broad spectrum 139 140## Quantum mechanics 141 142- uncertainty occurs in any measurement 143- inherent physical limit to absolute accuracy of measurements (result of wave-particle duality) 144- interaction between observer and object 145- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it 146 147### Indeterminancy principle 148 149$$\sigma E \sigma t \ge {h \over 4 \pi}$$ 150 151where $\sigma n$ is the uncertainty of $n$ 152 153**$\sigma E$ and $\sigma t$ are inversely proportional** 154 155Therefore, position and velocity cannot simultaneously be known with 100% certainty. 156 157### Single-slit diffraction 158 159- one photon passes through slit at any time (controlled by intensity) 160- diffraction pattern can be explained by wave front split into wavelets 161- diffraction can be represented as uncertainty of photonic momentum 162 163 164### Comparison with Bohr's model 165 166**Newtonian (deterministic) model** - current $x$ and $v$ are known, so future $x$ can be calculated 167 168**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 169 170Newton's and Einsteins models work together 171 172### Photon-electron interaction 173 174When a photon collides with an electron, momentum is transferred to electron. 175 176$$\rho_{\text{photon}}={h \over \lambda}$$ 177$$E=\rho c$$ 178 179 180 181 182 183774 abc melbourne