physics / light-matter-ref.mdon commit [methods] re-render methods notes (1dadb9e)
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   5author: Andrew Lorimer
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   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<!-- ![](graphics/energy-levels.png) -->
  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