physics / light-matter.mdon commit minor modifications to photoelectric notes (486c6cc)
   1# Light and matter
   2
   3## Photoelectric effect
   4
   5![](graphics/photoelectric-effect.png)
   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![](graphics/standing-wave-electrons.png)
 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
 115### Emission
 116
 117![](graphics/energy-levels.png)
 118
 119- Coloured lines show light being ejected from e- shells  
 120- Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$  
 121- Bohr's model describes discrete energy levels
 122- Energy is conserved (out = in)
 123- Ionisation energy - minimum energy required to remove an electron
 124- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
 125
 126## Light sources
 127
 128![](graphics/synchrotron.png)
 129
 130- **incandescent:** <10% efficient, broad spectrum
 131- **LED:** semiconducting doped-Si diodes
 132- - most electrons in *valence band* (energy level)
 133- - provided energy, electrons can jump to *conduction band* and move through Si as current
 134- - colour determined by $\Delta E$ between bands (shells), and type of doping
 135- **laser:** gas atoms are excited
 136- - *popular inversion* - most gas atoms are excited
 137- - photons are released if stimulated by another photon of the right wavelength
 138- **synchrotron:** - magnetically accelerates electrons
 139- - extremely bright
 140- - highly polarised
 141- - emitted in short pulses
 142- - broad spectrum
 143
 144## Quantum mechanics
 145
 146- uncertainty occurs in any measurement
 147- inherent physical limit to absolute accuracy of measurements (result of wave-particle duality)
 148- interaction between observer and object
 149- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it
 150
 151### Indeterminancy principle
 152
 153$$\sigma E \sigma t \ge {h \over 4 \pi}$$
 154
 155where $\sigma n$ is the uncertainty of $n$
 156
 157**$\sigma E$ and $\sigma t$ are inversely proportional**
 158
 159Therefore, position and velocity cannot simultaneously be known with 100% certainty.
 160
 161### Single-slit diffraction
 162
 163- one photon passes through slit at any time (controlled by intensity)
 164- diffraction pattern can be explained by wave front split into wavelets
 165- diffraction can be represented as uncertainty of photonic momentum
 166
 167
 168### Comparison with Bohr's model
 169
 170**Newtonian (deterministic) model** - current $x$ and $v$ are known, so future $x$ can be calculated
 171
 172**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
 173
 174
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