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