Planck-Einstein relation for photon energy:

$E = hf = \frac{hc}{\lambda}$

Where:

• h = 6.626 × 10-34 J - s (Planck's constant)
• f = frequency
• $\lambda$ = wavelength
• c = 3.00 × 108 m/s

Photon energy in eV: $E = \frac{1240\ \text{eV - nm}}{\lambda\ \text{(nm)}}$

Photons:

• Massless particles
• Travel at speed of light
• Quantized packets of electromagnetic energy
• Carry momentum: $p = \frac{h}{\lambda}$

Emission of electrons when light strikes metal surface.

Einstein's photoelectric equation: $K_{max} = hf - \phi$

Where:

• $K_{max}$ = maximum kinetic energy of ejected electrons
• hf = photon energy
• $\phi$ = work function (minimum energy to eject electron)

Key observations explained by photon theory:

• Threshold frequency exists ($f_c = \phi/h$)
• Below threshold, no electrons ejected regardless of intensity
• Above threshold, $K_{max}$ depends on frequency, not intensity
• Electron emission is instantaneous

Minimum energy required to remove electron from metal surface.

$\phi = hf_0$

Where $f_0$ = threshold frequency

Typical values (eV):

• Cesium: 2.1
• Sodium: 2.3
• Zinc: 4.3
• Platinum: 6.4

Higher work function: harder to eject electrons.

All matter exhibits both wave and particle properties.

Light:

• Wave: interference, diffraction
• Particle: photoelectric effect, photons

Matter:

• Particle: definite position, momentum
• Wave: diffraction patterns, interference

Complementary nature: both descriptions needed for complete understanding.

Wavelength associated with any moving particle:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

Where:

• h = Planck's constant
• p = momentum
• m = mass
• v = velocity

For electron (1 eV): $\lambda = 1.23\ \text{nm}$

Implications:

• Wavelength significant for small particles (electrons, atoms)
• Negligible for macroscopic objects
• Explains electron diffraction

Quantized model of hydrogen atom.

Postulates:

1. Electrons orbit at discrete radii
2. Only certain orbits allowed (angular momentum quantized)
3. Electron doesn't radiate in allowed orbit (stable)
4. Radiation emitted/absorbed when electron jumps between orbits

Quantization condition: $L = n\frac{h}{2\pi} = n\hbar$ (n = 1, 2, 3, ...)

Orbital radii: $r_n = n^2 a_0$

Where $a_0$ = Bohr radius = 5.29 × 10-11 m

Energy of electron in hydrogen atom (Bohr model):

$E_n = -\frac{13.6\ \text{eV}}{n^2}$

Where n = principal quantum number (1, 2, 3, ...)

Transitions:

• Electron absorbs photon (jumps to higher)
• Electron emits photon (drops to lower)
• Photon energy: $hf = E_f - E_i$

Series:

• Lyman: n -> 1 (UV)
• Balmer: n -> 2 (visible)
• Paschen: n -> 3 (infrared)

Emission spectrum:

• Light emitted when atoms de-excite
• Bright lines at specific wavelengths
• Each element has unique pattern

Absorption spectrum:

• Light absorbed when atoms excite
• Dark lines at specific wavelengths (continuous spectrum with gaps)
• Complementary to emission spectrum

Applications:

• Identifying elements (spectral analysis)
• Stars: continuous spectrum with absorption lines
• Neon signs: emission spectrum

Nucleus composed of protons and neutrons (nucleons).

Nuclear notation: $^A_Z X$

• Z = atomic number (number of protons)
• A = mass number (protons + neutrons)
• N = A - Z = number of neutrons

Properties:

• Extremely dense (~1017 kg/m3)
• Diameter: ~10-15 m (1 fm)
• Nuclear force: strong, short-range
• Electric force: repulsive between protons

Nuclei: same Z (same element, different neutrons)

Mass-Energy Equivalence E=mc2:

Einstein's famous relation between mass and energy:

$E = mc^2$

Where:

• E = energy (J)
• m = mass (kg)
• c = 3.00 × 108 m/s

Implies:

• Mass can be converted to energy
• Energy has mass equivalent
• Basis of nuclear reactions

In atomic units: $1\ \text{u} \times c^2 = 931.5\ \text{MeV}$

Where u = atomic mass unit = 1.66 × 10-27 kg

Mass defect: difference between mass of constituents and actual nucleus mass.

$\Delta m = (Z m_p + N m_n) - m_{nucleus}$

Binding energy: energy released when nucleus forms.

$E_b = \Delta m c^2$

Nuclear stability:

• Larger binding energy per nucleon = more stable
• Peak around iron (Fe-56)
• Fusion: light nuclei combine -> increase binding energy
• Fission: heavy nuclei split -> increase binding energy

Binding energy curve:

• Light nuclei: can fuse to become more stable
• Heavy nuclei: can fission to become more stable
• Iron: most stable (cannot release energy by either)

Alpha decay (alpha):

• Emits helium nucleus ($^4_2$He)
• Z decreases by 2, A decreases by 4
• Example: $^{238}_{92}\text{U} \rightarrow ^{234}_{90}\text{Th} + ^4_2\text{He}$
• High penetration shielding needed

Beta-minus decay (β-):

• Neutron converts to proton: $n \rightarrow p + e^- + \bar{\nu}_e$
• Z increases by 1, A unchanged
• Example: $^{14}_6\text{C} \rightarrow ^{14}_7\text{N} + e^- + \bar{\nu}_e$
• Moderate penetration, stopped by few mm of aluminum

Beta-plus decay (β+) (or electron capture):

• Proton converts to neutron: $p \rightarrow n + e^+ + \nu_e$
• Z decreases by 1, A unchanged
• Example: $^{11}_6\text{C} \rightarrow ^{11}_5\text{B} + e^+ + \nu_e$

Gamma decay (γ):

• Emits high-energy photon
• No change in Z or A
• Often follows other decays (de-excitation)
• Very high penetration, requires dense shielding (lead)

Time for half of radioactive nuclei to decay.

$t_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$

Where $\lambda$ = decay constant

Number remaining after time t: $N(t) = N_0 e^{-\lambda t} = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}$

Activity (decays per second): $A(t) = \lambda N(t) = A_0 e^{-\lambda t}$

Applications:

• Radiometric dating (carbon-14)
• Medical imaging (radioactive tracers)
• Nuclear medicine treatment

Nuclear fission:

• Heavy nucleus splits into lighter fragments
• Releases neutrons + energy
• Example: $^{235}_{92}\text{U} + n \rightarrow ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3n + \text{energy}$
• Chain reaction possible (neutrons trigger more fissions)
• Used in nuclear power plants and weapons

Nuclear fusion:

• Light nuclei combine to form heavier nucleus
• Requires extremely high temperature/pressure
• Example: $^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + n + \text{energy}$
• Powers stars (hydrogen fusion in Sun)
• Promising for future energy (fusion reactors)

Energy comparison:

• Fusion per nucleon: ~3-4 times more energy than fission
• Both much more energy dense than chemical reactions