Biophysical Spectroscopy Atlas

Thirty-two techniques arranged by the physical process they exploit, each with its governing equations, what it reveals about biomolecules, and where it stops working. Select any technique, on the map or on the spectrum, to open its full card. Four live calculators and a question-to-method guide close the page.

Level 1Physical processabsorption, emission, scattering, resonance, time-resolved and nonlinear

Level 2Transition or observableelectronic, vibrational, spin, polarization, correlation

Level 3TechniqueCD, FTIR, FRET, NMR, DEER, SAXS and others

Level 4Variants and methodspulse sequences, labels, geometries and analysis inside each card

Energy & units

Photon energy

\[E = h\nu = \frac{hc}{\lambda} = hc\,\tilde\nu\]

1 eV = 8065.5 cm⁻¹ = 241.8 THz = 96.49 kJ mol⁻¹. Thermal energy at 298 K: k_BT = 207 cm⁻¹ = 2.48 kJ mol⁻¹ = 25.7 meV. Transitions far above k_BT start from the ground state; those below it are thermally populated.

Transition rates

Fermi's golden rule

\[w_{i\to f} = \frac{2\pi}{\hbar}\,\bigl|\langle f|\hat H'|i\rangle\bigr|^2\,\rho(E_f)\]

For electric-dipole coupling Ĥ′ = −μ̂·E(t). Every absorption, emission and scattering rate descends from this: squared coupling matrix element times density of final states.

Intensity & polarization

Transition dipole and selection rules

\[\boldsymbol\mu_{fi} = \langle\psi_f|\hat{\boldsymbol\mu}|\psi_i\rangle,\qquad P\propto|\mathbf E\cdot\boldsymbol\mu_{fi}|^2\propto\cos^2\theta\]

Electronic: Δl = ±1, ΔS = 0. Vibrational: Δv = ±1 with ∂μ/∂Q ≠ 0 (IR) or ∂α/∂Q ≠ 0 (Raman); mutual exclusion in centrosymmetric molecules. Spin: Δm_I = ±1 (NMR), Δm_S = ±1 (EPR). The cos²θ law underlies LD, anisotropy and κ².

Absorption ↔ emission

Einstein coefficients and oscillator strength

\[A_{21} = \frac{8\pi h\nu^3}{c^3}\,B_{21},\qquad f\approx 4.32\times10^{-9}\!\int\!\varepsilon(\tilde\nu)\,d\tilde\nu\]

Spontaneous emission scales with ν³: radiative decay dominates in the UV–visible (ns) but is negligible for spin transitions, where relaxation takes over. f is dimensionless (ε in M⁻¹cm⁻¹, ν̃ in cm⁻¹); strong allowed bands have f ≈ 0.1–1.

Quantitation

Beer–Lambert law

\[A=\log_{10}\frac{I_0}{I}=\varepsilon\,c\,l,\qquad \sigma\approx3.82\times10^{-21}\,\varepsilon\ \ \mathrm{cm^2}\]

Linear for dilute, non-scattering, non-interacting absorbers. Curvature flags aggregation, stray light, scattering or saturation. σ is the molecular absorption cross-section.

Line shapes

Homogeneous and inhomogeneous broadening

\[L(\nu)=\frac{1}{\pi}\frac{\Gamma/2}{(\nu-\nu_0)^2+(\Gamma/2)^2},\quad \Gamma=\frac{1}{\pi T_2}\]

\[G(\nu)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-(\nu-\nu_0)^2/2\sigma^2},\quad \mathrm{FWHM}=2\sqrt{2\ln2}\,\sigma\]

Dephasing and lifetime give Lorentzians (Δν = 1/2πτ: a 1 ns lifetime gives only 0.005 cm⁻¹). A static spread of environments gives Gaussians. Real bands are Voigt convolutions; 2D spectroscopy and hole burning separate the two.

Vibronic structure

Franck–Condon principle

\[I_{0\to n}\propto\bigl|\langle\chi'_n|\chi''_0\rangle\bigr|^2 = e^{-S}\frac{S^n}{n!}\]

Nuclei are frozen during an electronic transition. The Huang–Rhys factor S sets the vibronic progression; the Stokes shift ≈ 2Sħω is twice the reorganization energy (Marcus λ). Explains the absorption–emission mirror image.

Sensitivity

Boltzmann populations

\[\frac{N_\text{upper}}{N_\text{lower}}=e^{-\Delta E/k_BT},\qquad P\approx\frac{\Delta E}{2k_BT}\]

Net polarization P at 298 K: ¹H at 600 MHz ≈ 5×10⁻⁵; electrons at 9.5 GHz ≈ 8×10⁻⁴; electronic transitions ≈ 1. This is why NMR needs mM samples, EPR µM, and fluorescence pM. Hyperpolarization (DNP) breaks the limit.

Causality

Kramers–Kronig relations

\[\chi'(\omega)=\frac{2}{\pi}\,\mathcal{P}\!\!\int_0^\infty\!\frac{\omega'\,\chi''(\omega')}{\omega'^2-\omega^2}\,d\omega'\]

Absorption (χ″) and dispersion (χ′, refractive index) are two faces of one response. CD ↔ ORD, absorption ↔ refractive-index shifts in SPR, absorptive ↔ dispersive NMR phases.

Relaxation

Bloch equations

\[\frac{d\mathbf M}{dt}=\gamma\,\mathbf M\times\mathbf B-\frac{M_x\hat{\mathbf x}+M_y\hat{\mathbf y}}{T_2}-\frac{(M_z-M_0)\,\hat{\mathbf z}}{T_1}\]

T₁ restores populations; T₂ destroys coherence, with 1/T₂ = 1/2T₁ + 1/T₂′. The optical Bloch equations have the same form for two-level electronic systems, which is why pulse NMR, pulse EPR and nonlinear optics share one vocabulary.

Measurement

Fourier-transform spectroscopy

\[S(\omega)=\int_{-\infty}^{\infty}s(t)\,e^{-i\omega t}\,dt,\qquad \Delta\tilde\nu\approx\frac{1}{L_\mathrm{OPD}}\]

Record a time-domain signal (NMR free induction decay, FTIR interferogram, pulse-EPR echo, 2D coherence times) and transform. Gains: multiplex (Fellgett) and throughput (Jacquinot). Resolution is set by acquisition length.

Unifying idea

Spectra are Fourier transforms of correlation functions

\[I(\omega)\propto\int_{-\infty}^{\infty}\langle\boldsymbol\mu(0)\cdot\boldsymbol\mu(t)\rangle\,e^{-i\omega t}\,dt\]

Line shapes come from dipole correlations, NMR relaxation from spectral densities J(ω), and DLS, FCS and anisotropy decays measure correlation functions directly. Fluctuation–dissipation links them all to equilibrium dynamics.

Biophysical Spectroscopy Atlas

Six process families, twenty-two observables, thirty-two techniques

Where each technique sits on the electromagnetic spectrum

Twelve results every technique rests on

Photon energy

Fermi's golden rule

Transition dipole and selection rules

Einstein coefficients and oscillator strength

Beer–Lambert law

Homogeneous and inhomogeneous broadening

Franck–Condon principle

Boltzmann populations

Kramers–Kronig relations

Bloch equations

Fourier-transform spectroscopy

Spectra are Fourier transforms of correlation functions

Which technique sees which motion

Four numbers you will compute again and again

Energy and wavelength converter

Efficiency versus distance

Lorentzian, Gaussian and Voigt

From molar mass to diffusion and tumbling

Which method answers my question?

Standard references