1--- 2geometry: margin=2cm 3columns: 2 4graphics: yes 5author: Andrew Lorimer 6linestretch: 0.9 7--- 8 9\pagenumbering{gobble} 10<!-- \usepackage{multicols} --> 11 12# Light and Matter 13 14## Planck's equation 15 16$$f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c$$ 17 18$$h=6.63\times 10^{-34}\operatorname{J s}=4.14\times 10^{-15} \operatorname{eV s}$$ 19 20$$ 1 \operatorname{eV} = 1.6\times 10^{-19} \operatorname{J}$$ 21 22## Force of electrons 23 24$$F={2P_{\text{in}}\over c}$$ 25 26$$\text{photons per second}={\text{total energy} \over \text{energy per photon}}={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}$$ 27 28## Photoelectric effect 29 30- $V_{\operatorname{supply}}$ does not affect photocurrent 31- $V_{\operatorname{sup}} > 0$: e- attracted to collector anode 32- $V_{\operatorname{sup}} < 0$: attracted to illuminated cathode, $I\rightarrow 0$ 33- $v$ of e- depends on ionisation energy (shell) 34- max current depends on intensity 35 36### Threshold frequency $f_0$ 37- minimum $f$ for photoelectrons to be ejected 38- $x$-intercept of frequency vs $E_K$ graph 39- if $f < f_0$, no photoelectrons are detected 40 41### Work function $\phi$ 42- minimum $E$ required to release photoelectrons 43- magnitude of $y$-intercept of frequency vs $E_K$ graph 44- $\phi$ is determined by strength of bonding 45 46$$\phi=hf_0$$ 47 48### Kinetic energy 49 50$$E_{\operatorname{k-max}}=hf - \phi$$ 51 52voltage in circuit or stopping voltage = max $E_K$ in eV 53equal to $x$-intercept of volts vs current graph (in eV) 54 55### Stopping potential ($V$ for minimum $I$) 56 57<!-- $$V_0 = {E_{K \operatorname{max}} \over q_e} = {{hf - \phi} \over q_e}$$ --> 58$$V=h_{\text{eV}}(f-f_0)$$ 59 60## De Broglie's theory 61 62$$\lambda = {h \over \rho} = {h \over mv}$$ 63$$\rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c$$ 64 65- cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$) 66<!-- - confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit) --> 67- confirmed by similar e- and x-ray diff patterns 68 69## X-ray and electron interaction 70 71- e- is only stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ 72- rearranging this, $2\pi r = n{h \over mv} = n \lambda$ (circumference) 73- if $2\pi r \ne n{h \over mv}$, no standing wave 74- if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$ 75- calculating $h$: $\lambda = {h \over \rho}$ 76 77## Spectral analysis 78 79<!--  --> 80- $\Delta E = hf = {hc \over \lambda}$ between ground / excited state 81- $E$ and $f$ of photon: $E_2 - E_1 = hf = hc$ 82- Ionisation energy - min $E$ required to remove e- 83- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$) 84 85## Indeterminancy principle 86 87measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it. 88 <!-- $\therefore, \sigma E \propto {1 \over \sigma t}$ --> 89 90<!-- $$\sigma E \sigma t \ge {h \over 4 \pi}$$ --> 91 92$$\sigma \rho \sigma x = {h \over 4\pi}$$ 93 94## Wave-particle duality 95wave model: 96 97- cannot explain photoelectric effect 98- $f$ is irrelevant to photocurrent 99- predicts delay between incidence and ejection 100- speed depends on medium 101 102particle model: 103 104- explains photoelectric effect 105- rate of photoelectron release $\propto$ intensity 106- no time delay - one photon releases one electron 107- double slit: photons interact. interference pattern still appears when a dim light source is used so that only one photon can pass at a time 108- light exerts force 109- light bent by gravity