Photoelectric Effect: Key Equations and Graphs

Photoelectric Effect

• Kinetic energy, K.E., of an emitted photoelectron equals the elementary charge, e, times the stopping potential, V₀: K.E. = eV₀
• Intensity (I) is the measure of power per unit area; can also be calculated from E = nhν when n is the number of photons
• Einstein's theory stipulates energy is quantized (E = nhν) and energy absorption/emission occurs discretely
• A photon's energy (E) equals the maximum kinetic energy of emitted electrons plus the work function (Φ): E = (K.E.)max + Φ
• The work function is equal to Φ = hν₀, where ν₀ is threshold frequency
• The equation eV₀ = hν - hν₀ relates stopping potential to frequency and threshold frequency
• V₀ = (ℎ/𝑒) ν - (ℎν₀/𝑒) dictates the relationship between stopping potential and frequency
• Slope = m = h/e, this is material-independent
• Y-intercept = c = - hν₀/e, this this material-dependent

Important Graphs

• Photocurrent (Ip) increases with intensity (I); Photocurrent is directly proportional to intensity, I ∝ Ip
• With the same frequency but different intensity, same frequency emits electrons with same kinetic energy, the same stopping potential
• Higher intensity produces more photocurrent when I₂ > I₁
• Photocurrent is zero when ν = 0; Ip ≠ 0
• Two beams with the same intensity but different frequency, the greater the frequency, the greater kinetic energy and stopping potential
• ν₁ > ν₂

Compton Effect

• Scattered photons posses less energy i.e. E'↑, f ↓, λ↑ compared to incident photons
• Describes the change in wavelength of X-rays when scattered by matter using the formula
• Recoiled electron from photoelectric effect is non-relativistic i.e. KE <<< mc²
• Compton shift, independent of wavelength, is quantified by Δλ = h/(m₀c) [1 - cos∅]
• Wavelength after scattering is λ' = λ + h/(m₀c)[1 - cos∅] = λ + λc [1 − cosф]
• Compton wavelength, λc = h/(m₀c) = 0.02426 Å
• When φ = 0, then Δλ = 0 and λ' = λ
• When φ = 180, then Δλ = λmaximum and λ' = λ + 2h/(m₀c)
• Relation between θ and φ: cotθ = tan(φ/2) [1 + (ℎν)/(m₀c²)]
• Recoiled electron kinetic energy is K.E. = E - E' = hv - hv' ≈ hc(1/λ - 1/λ’)

Difference Between Photoelectric and Compton Effects

• Photoelectric Effect: low energy phenomena, Compton Effect: moderate energy phenomena
• Photoelectric Effect: electron is bounded (effective heavy mass); Compton Effect: electron is free (effective light mass)
• Photoelectric Effect: photon completely transfers energy to one electron; Compton effect: photon transfers a some amount of energy to one electron
• In Compton effect, if photon transfers complete energy, conservation of momentum would be violated

De Broglie Hypothesis

• States that every particle has a wavelength related to its momentum: λ = h/p = h/mv
• For a single plane matter wave: y = A sin(ωt - kx)
• For a group of plane matter waves: y₁ = Asin(kx-ωt), y₂ = Asin[(k+∆k)x-(ω+Δω)t]
• Resultant wave, y = 2Asin(kx-ωt)cos ((∆kx - ∆ωt)/2)
• Phase velocity: vp = ω/k
• Group velocity: vg = ∆ω/∆k = dω/dk
• vpvg = c²

Heisenberg Uncertainty Principle

• Δx ⋅ Δp ≥ ℏ/2
• ΔE ⋅ Δt ≥ ℏ/2

Single Slit Diffraction

• From conservation of momentum, Δpy = 2psinθ
• For Minima, dsinθ ⋅nλ ⇒d = ⁿλ/sinθ = ⁿℏ/psing, where n = order of minima
• ∆y ⋅ ∆py = d(2psinθ) = 2nℏ
• For n = 1, ∆y ⋅ ∆py ≥ 2ℏ ≈ ∆y ⋅ ∆py ≥ 4πℏ (just after passing through the slit)

Blackbody Radiations

• Various terms are used to describe aspects of blackbody radiation
• uλ: Spectral energy density, the measure of energy per unit volume per unit wavelength λ
• uλ dλ represents the energy per unit volume within the wavelength range λ and λ + dλ
• u: Total energy density, the total radiant energy for all wavelengths from 0 to ∞ per unit volume
• Total spectral energy is related density by u = ∫₀^∞ uλ dλ
• eλ: Spectral emissive power, the radiant energy per second per unit surface area per unit range of λ
• eλ dλ describes the energy per unit area per second within the wavelength range λ and dλ
• e: Emissivity, total emissive power or power per unit area
• It is defined as the total radiant energy of all wavelengths from 0 to ∞ per second per unit area
• The total density is related to spectral emissive power by e = ∫₀^∞ eλ dλ
• aλ: Spectral absorptivity, the fraction of incident energy absorbed per unit surface area per second at λ
• For a normal body, the probability of reflection (rλ), absorption (aλ), and transmission (tλ) equals 1: rλ + aλ + tλ = 1
• A black body absorbs all radiation, thus rλ = tλ = 0, and aλ = 1
• For EM waves, radiation pressure P = I/c where I = intensity of radiation
• In a container, P = u/3, with u representing energy density
• Kirchhoff's law states that for a perfect black-body, Eλ/aλ = constant
• This constant implies a good absorber of radiation is also a good emitter and vice-versa
• Stefan's Law dictatates that the energy density is proportional to the fourth power of temperature
• u ∝ T⁴ and u = σT⁴, with σ as a constant.
• The total rate of emission of radiant energy per unit area is proportional to the energy density; power/area ∝ F ∝ u
• For Power = E x Area = σT × Area where sigma = 5.672 x 10⁻⁸ J/m²/k⁴
• For Emissive power = E = σT⁴ / Area
• Given a black-body is in an enclosure at temperature T₀, its emission E = σ(T⁴ - T₀⁴)

Important Relations

• Radiation Pressure: P = u/3 = aT⁴/3
• Total Energy: U = u × volume = aT⁴V
• Specific Heat: For costant volume, Cv = (dU/dT)ᵥ = 4atT³
• Change in entropy: at a constant volume. if temp. changes from 0 to T then *∆S= 4/3 * aV³
• Helmoholtz Free Energy: F = U - TAS = -1/3 aVT⁴
• Gibbs Free Energy: G = F + PV = 0 (for BBR)

Rayleigh-Jeans Law

• Describes the spectral energy density in terms of frequency: uᵥdv = (8πk(B)T/c³)v²dv
• Describes the spectral energy density in terms of wavelength: |uλdλ| = (8πk(B)T/λ⁴)dλ
• The law implies uλ → ∞ as λ → 0 or ν → ∞, a phenomenon known as the ultraviolet catastrophe

Wien's Law

• uλ dλ = aλ⁻⁵exp(-b/λT) dλ
• The area under the curve, represented by μ = ∫₀^∞ uλdλ, increases as µ or T increases
• λm stands for the value of wavelength at which spectral energy density reaches its maximum
• With increasing temperature, the spectral density and the value of am decreases
• At a particular value, with Wien's displacement law: λT = constant = b= 2.8 x 10^-3 mK
• ⇒λm₁₁ = λm₂T₂

Quantum Mechanics - Probability Density

• ψ² = ψ*ψ
• Represents the probablity density (ψ) depends on position and time.
• Probability of a particle existing is represented by ∫₀^∞ |ψ (x)²| = 1
• Dimentions of a wave function for Linear [[ψ]] = 𝐿-½, Area [[ψ]] = 𝐿⁻¹, Volume [[ψ]] = 𝐿⁻³/²

Quantum Mechanics - Allowed Wave Function

• Wave function is finite and continous.
• For bound wave function ψ(x = +∞) = 0
• First order derivatives may be continous or discontinous depending on the potential.
• (∂ψ/∂x) is continous if potential at the boundary has finite discontinutity
• (∂ψ/∂x) is discontinous if potential at the boundary has infinite discontinutity
• Wave function must be square integrable ∫₀^∞ *ψ^ψ = 1 Normalisation condition
• Wave function must be single valued

Quantum Mechanics - Gamma Integral

• TYPE ONE ∫₀^∞ X^(n-1) E^(-ax) dx = Γ(n)/α^n = ((n-1)!)/α^n
• TYPE TWO ∫₀^∞ X^(2n) E^(-a(X^2)) dx = ((n-1)!)/(2a^((n+1)/2))

Quantum Mechanics - Operators

• Âψ = λψ For same state, ψ is an eigenstate λ is an eigenvalue
• Âψ = φ For different state, ψ is not an eigenstate only transformation occurs

Quantum Mechanics - Properties of Operators

• Position operator (𝑥): 𝑥ψ = 𝑥ψ
• Momentum operator (𝑝): 𝑝 = −ih(∂/∂x)
• In 3D p = px(i) + py(j) + pz(k)
• Where (p x) = −ih(∂/∂x), (p y) = −ih(∂/∂y), (p z) = −ih(∂/∂z)
• Kinetic energy operator (𝑘): 𝑘 = ((p𝑥^2)/(2m)) = −ħ²/2𝑚 (∂²/∂x²),
• Hamiltonian operator (𝐻): Gives the total energy of the system: 𝐻ψ = 𝐸ψ