Фізика молекул та міжмолекулярних взаємодій

🌊 Plasma oscillations

Plasmon frequency

ω_p = √(ne²/(ε₀m)). Collective excitations. Metals: ω_p ~ 10¹⁶ рад/с (visible/UV).

Dielectric function

ε(ω) = 1 - ω_p²/ω². Re(ε) < 0 для ω < ω_p. Reflectivity, transmission.

Debye screening

λ_D = √(ε₀k_B T/(ne²)): screening length. Static screening, electrostatic shielding.

Applications

Plasmonics, surface plasmons, optics. Sensors, nano-optics.

⚛️ Electron-phonon

Fröhlich Hamiltonian

H_int = Σ M_q(a_q† + a_{-q}) c_{k+q}† c_k. Coupling: phonon emission/absorption.

Coupling constant

λ: dimensionless. BCS: λ < 1 weak, λ > 1 strong. McMillan formula.

Superconductivity

BCS: Cooper pairing via phonons. T_c ~ exp(-1/λ). Eliashberg theory.

Applications

Superconductors, transport. Resistivity, mobility. Polarons.

🔬 Polarons

Large polaron

Continuum: Pekar. m* > m, mobility reduced. Optical absorption broadening.

Small polaron

Atomic limit: localized. Strong coupling. Holstein model. Self-trapping.

Effective mass

m*/m ~ 1+α/6 для large polaron. α: coupling strength. Enhancement.

Transport

Mobility μ ∝ exp(-W/(k_B T)) для small polarons. Activation energy W.

🌊 Many-body correlations

Hubbard model

H = -t Σ c_i†c_j + U Σ n_i↑n_i↓. Metal-insulator transition: U/t.

Quantum criticality T = 0 phase transitions. Critical exponents, scaling. Correlation length divergence.

T = 0 phase transitions. Critical exponents, scaling. Correlation length divergence.

Luttinger liquid

1D systems: breakdown Fermi liquid. Power law correlations. Charge, spin separation.

Non-Fermi liquid

Strong correlations. Marginal Fermi liquid, strange metals. Anomalous behavior.

🔬 Collective excitations

Plasmons

ω ~ √k для surface plasmons. Dispersion relations. Losses γ, lifetime τ.

Phonons

Lattice vibrations: longitudinal, transverse. Dispersion ω(k). Debye model.

Magnons

Spin waves: ω(k) ~ Dk². GHz frequencies. Quantized magnetic excitations.

Landau damping

Mode decay: resonant particle-wave interaction. Critical velocity, dissipation.

🌊 Screening effects

Thomas-Fermi

Static screening: exponential. λ_TF ~ 1/√(g(E_F)). Metallic systems.

Lindhard

Dynamic dielectric ε(k,ω): frequency-dependent. Collective modes, particle-hole.

Random Phase

RPA: dielectric approximation. Plasmon dispersion. Screening cloud.

Applications

Electron systems, plasmas. Transport, optical response.

📊 Графіки та діаграми

Plasma dispersion

ω(k):

Collective mode: plasmon. Gap at ω_p, linear @ high k. Damping.

Polaron binding

E_p:

Self-energy: attractive. Effective mass m* enhanced. Mobility.

ω² = ω_p² + 3/5·v_F²·k² Де ω_p = √(ne²/(ε₀m)) Bohm-Gross dispersion Landau damping @ high k

🧪 Практичні приклади

Приклад 1: Metal plasma

Al: ω_p ~ 15 eV (~10¹⁶ рад/с). Reflectivity high @ ω < ω_p. Optics.

Приклад 2: BCS superconductor

Al: T_c ~ 1.2 K, λ ~ 0.4. Phonon-mediated pairing. McMillan formula.

Приклад 3: Mott transition

V₂O₃: U/t ~ 10. Insulator → metal. Temperature, pressure driven.

Приклад 4: Polaronic transport

Organic semiconductors: small polarons. Mobility μ ∝ exp(-W/(k_B T)).

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Молекули: флуктуації та кореляції

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