Microscopy & spectroscopy formulas

$d$ = minimum resolvable distance (same unit as $\lambda$); $\lambda$ = wavelength of the illumination; $\mathrm{NA}=n\sin \theta$ = numerical aperture of the objective, where $n$ is the refractive index of the medium between specimen and objective (air $\approx 1.00$, immersion oil $\approx 1.52$) and $\theta$ is the half-angle of the cone of light the objective accepts. Smaller $d$ = better resolution. Shorter wavelength and/or higher NA improve resolution; oil immersion raises NA (to about 1.4–1.5).

Abbe's form gives a slightly smaller $d$ than the Rayleigh form for the same $\lambda$ and NA. The two are different criteria, not different conventions for the same number — use the form the question specifies.

$h$ = Planck's constant; $\nu$ = frequency; $c$ = speed of light; $\lambda$ = wavelength; $\tilde{\nu }=1/\lambda$ = wavenumber (usually in $\mathrm{c}{\mathrm{m}}^{-1}$). Shorter wavelength = higher energy.

$I_0$ = intensity of light entering the sample; $I$ = intensity transmitted; $T$ = transmittance. Absorbance is dimensionless. Example: $T=10\%$ gives $A=1$; $T=1\%$ gives $A=2$.

$A$ = absorbance (dimensionless); $ϵ$ = molar absorption (extinction) coefficient in ${\mathrm{M}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$, specific to the molecule, wavelength and solvent; $c$ = molar concentration (M = mol ${\mathrm{L}}^{-1}$); $l$ = path length (cm; a standard cuvette is 1 cm). Holds for monochromatic light in dilute, non-scattering solutions, within the linear range of the instrument — at high absorbance (often above about 1) the response deviates from linearity and the sample should be diluted. For a mixture of non-interacting absorbers, absorbances at a given wavelength add.

Conventional factors $F$: an $A_{260}$ of 1.0 in a 1 cm path corresponds to about 50 µg/mL double-stranded DNA, about 40 µg/mL RNA, and about 33 µg/mL single-stranded DNA.

A rough purity indicator only. A ratio well below these values suggests protein (or phenol) contamination; a DNA preparation reading near 2.0 may contain RNA. The ratio is affected by the pH of the solution. A companion check, $A_{260}/A_{230}$, is about 2.0–2.2 for clean nucleic acid; lower values indicate carry-over of contaminants that absorb near 230 nm (e.g. guanidine salts, phenol, carbohydrates).

$E$ = efficiency of energy transfer from donor to acceptor; $r$ = donor–acceptor distance; $R_0$ = Förster distance, the distance at which $E=0.5$ (typically a few nanometres). Requires overlap of the donor emission spectrum with the acceptor excitation spectrum. The sixth-power dependence makes FRET a sensitive 'molecular ruler' over roughly 1–10 nm.

$A_{\mathrm{L}}$, $A_{\mathrm{R}}$ = absorbance of left- and right-circularly polarised light. $\Delta A$ is non-zero only for chiral chromophores, or chromophores in a chiral environment (e.g. peptide bonds in a folded protein, bases in a nucleic-acid helix). CD is often reported as ellipticity.

$z$ = charge number; $e$ = elementary charge; $V$ = accelerating potential; $m$ = ion mass; $v$ = velocity in the field-free flight tube; $L$ = flight-tube length; $t$ = flight time. Every ion leaves the source with kinetic energy $zeV$, so for the same $V$ and $L$: at a fixed charge a heavier ion arrives later; at a fixed mass a more highly charged ion arrives sooner.

$M$ = neutral molecular mass (Da); $n$ = number of protons added (charge); $m_{{\mathrm{H}}^{+}}$ = mass of a proton $\approx 1.007$ Da (often rounded to 1). For two ADJACENT peaks $p_1<p_2$ in the charge-state series, $p_2$ carries charge $n$ (given by the second formula) and $p_1$ carries $n+1$; then $M=n(p_2-m_{{\mathrm{H}}^{+}})$.

$\nu$ = resonance (Larmor) frequency; $\gamma$ = gyromagnetic ratio of the nucleus; $B_0$ = applied magnetic field. The chemical shift $\delta$ is measured relative to a reference (tetramethylsilane, TMS, set at $\delta =0$) and is independent of field strength. A stronger field spreads the signals further apart (better resolution and sensitivity).