## Photoelectric effect
+
+
### Planck's equation
-$$E=hf$$
+$$E=hf,\quad f={c \over \lambda}$$
+$$\therefore E={hc \over \lambda}$$
where
$E$ is energy of a quantum of light (J)
-$f$ is frequency of EM radiation
-$h$ is Planck's constant ($6.63 \times 10^{-34}\operatorname{J s}$)
+$f$ is frequency of EM radiation
+$h$ is Planck's constant ($6.63 \times 10^{-34}\operatorname{J s}=4.12 \times 10^{-15} \operatorname{eV s}$)
+
+### Electron-volts
+
+$$ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$$
+
+*Amount of energy an electron gains when it moves through a potential difference of 1V*
+
+- equivalent unit is Joule seconds (e.g. $h$)
+
+### Photoelectric effect
+
+- some metals becomes positively charged when hit with EM radiation
+- this is due to e- being ejected from surface of metal
+- *photocurrent* - flow of e- due to photoelectric effect
+- causes increase in current in a circuit
+- $V_{\operatorname{supply}}$ does not affect photocurrent
+- if $V_{\operatorname{supply}} \gt 0$, e- are attracted to collector anode.
+- if $V_{\operatorname{supply}} \lt 0$, e- are attracted to illuminated cathode, and $I\rightarrow 0$
+- not all electrons have the same velocity - depends on ionisation energy (shell)
+
+#### Wave / particle (quantum) models
+wave model:
+
+- cannot explain photoelectric effect
+- $f$ is irrelevant to photocurrent
+- predicts that there should be a delay between incidence of radiation and ejection of e-
+
+particle model:
+
+- explains photoelectric effect
+- rate of photoelectron release is proportional to intensity of incident light
+- shining light on a metal "bombards" it with photons
+- no time delay
+- one photon releases one electron
+
+#### Work function and threshold frequency
+
+- *threshold frequency* $f_0$ - minimum frequency for photoelectrons to be ejected
+- if $f \lt f_0$, no photoelectrons are detected
+
+- Einstein: energy required to eject photoelectron is constant for each metal
+- *work function* $\phi$ - minimum energy required to release photoelectrons
+- $\phi$ is determined by strength of bonding
+
+$$\phi=hf_0$$
+
+#### $E_K$ of photoelectrons (stopping energy)
+
+$$E_{\operatorname{k-max}}=hf - \phi$$
+
+where
+$E_k$ is max energy of an emmitted photoelectron
+$f$ is frequency of incident photon (**not** emitted electron)
+$\phi$ is work function ("latent" energy)
+
+Gradient of a frequency-energy graph is equal to $h$
+y-intercept is equal to $\phi$
+voltage $V$ in circuit is indicative of max kinetic energy in eV
+
+#### Stopping potential $V_0$
+
+Smallest voltage to achieve minimum current
+
+$$V_0 = {E_{K \operatorname{max}} \over q_e} = {{hf - \phi} \over q_e}$$
+
+## Wave-particle duality
+
+### Double slit experiment
+Particle 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.
+
+## De Broglie's theory
+
+$$\lambda = {h \over \rho} = {h \over mv}$$
+
+- theorised that matter may display both wave- and particle-like properties like light
+- predict wavelength of a particle with $\lambda = {h \over \rho}$ where $\rho = mv$
+- 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 ($< r_{\operatorname{proton}}$)
+- confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit)
+- also confirmed by Thomson - e- diffraction pattern resembles x-ray (wave) pattern
+- electron is only stable in orbit if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
+- rearranging this, $2\pi r = n{h \over mv}$ (circumference)
+- therefore, stable orbits are those where circumference = whole number of e- wavelengths
+- if $2\pi r \ne n{h \over mv}$, interference occurs when pattern is looped and standing wave cannot be established
+
+
+
+### Photon momentum
+
+$$\rho = {hf \over c} = {h \over \lambda}$$
+- if a massy particle (e.g. electron) has a wavelength, then anything with a wavelength must have momentum
+- therefore photons have (theoretical) momentum
+- to solve photon momentum, rearrange $\lambda = {h \over mv}$
+
+## Spectral analysis
+
+
+### Absorption
+- Black lines in spectrum show light not reflected
+- Frequency of a photon emitted or absorbed can be calculated from energy difference: $E_2 – E_1 = hf$ or $= hc$
+
+### Emission
+
+
+
+- Coloured lines show light being ejected from e- shells
+- Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$
+- Bohr's model describes discrete energy levels
+- Energy is conserved (out = in)
+- Ionisation energy - minimum energy required to remove an electron
+- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
+
+## Light sources
+
+
+
+- **incandescent:** <10% efficient, broad spectrum
+- **LED:** semiconducting doped-Si diodes
+- - most electrons in *valence band* (energy level)
+- - provided energy, electrons can jump to *conduction band* and move through Si as current
+- - colour determined by $\Delta E$ between bands (shells), and type of doping
+- **laser:** gas atoms are excited
+- - *popular inversion* - most gas atoms are excited
+- - photons are released if stimulated by another photon of the right wavelength
+- **synchrotron:** - magnetically accelerates electrons
+- - extremely bright
+- - highly polarised
+- - emitted in short pulses
+- - broad spectrum
+
+## Quantum mechanics
+
+- uncertainty occurs in any measurement
+- inherent physical limit to absolute accuracy of measurements (result of wave-particle duality)
+- interaction between observer and object
+- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it
+
+### Indeterminancy principle
+
+$$\sigma E \sigma t \ge {h \over 4 \pi}$$
+
+where $\sigma n$ is the uncertainty of $n$
+
+**$\sigma E$ and $\sigma t$ are inversely proportional**
+
+Therefore, position and velocity cannot simultaneously be known with 100% certainty.
+
+### Single-slit diffraction
+
+- one photon passes through slit at any time (controlled by intensity)
+- diffraction pattern can be explained by wave front split into wavelets
+- diffraction can be represented as uncertainty of photonic momentum
+
+
+### Comparison with Bohr's model
+
+**Newtonian (deterministic) model** - current $x$ and $v$ are known, so future $x$ can be calculated
+
+**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
+
+Newton's and Einsteins models work together
+### Photon-electron interaction
-### Electron diffraction patterns
+When a photon collides with an electron, momentum is transferred to electron.
-$$W=qV$$
+$$\rho_{\text{photon}}={h \over \lambda}$$
+$$E=\rho c$$
-(work for accelerating electon of charge $q$ with voltage $V$)
-$$\lambda = {h \over mv}$$
-(de Broglie's equation)
-Solving wavelength of electrons from gun:
-1.
774 abc melbourne
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