Spectroscopy Nomenclature

Spectroscopic results are often numerically correct but semantically incomplete. A value called a wavelength may be a vacuum wavelength, an air wavelength, or a wavelength inside a sample. An absorbance may use a base-10 or natural logarithm. A quoted linewidth may be a full width at half maximum, a half width, a Gaussian standard deviation, or a decay rate. These are not typographical details: changing the convention changes the number and sometimes the physical interpretation.

This page is a compact vocabulary and conversion reference. It states enough of each definition to audit a table, spectrum, or methods section. The canonical derivations remain in Constants and Conversions, Absorption and Emission, and Line Shape Reference.

Canonical Scope

This entry owns:

• vacuum wavelength, medium wavelength, ordinary frequency, angular frequency, spectroscopic wavenumber, and angular wavenumber conventions;
• transmittance, decadic and Napierian absorbance, absorptance, attenuation, and optical depth;
• microscopic cross sections and their connection to macroscopic coefficients;
• the minimum metadata needed to interpret a linewidth;
• elementary diatomic branch notation and warnings for open-shell spectra; and
• a reporting checklist for reproducible spectra.

It does not own:

• numerical values of fundamental constants;
• detailed line-profile theory or broadening mechanisms;
• transition moments, oscillator strengths, or Einstein coefficients;
• radiative-transfer solutions with distributed emission;
• molecular term-symbol construction; or
• a full assignment theory for polyatomic, asymmetric-top, or predissociative spectra.

Follow the linked canonical pages when those details matter.

Symbol-Collision Ledger

No symbol is self-defining. The following collisions are common enough that a paper or database should resolve them in words.

Symbol	Common meanings	Safer practice
$A$	decadic absorbance; Einstein spontaneous-emission coefficient; area	write $A_{10}$ for decadic absorbance and $A_{ul}$ for an Einstein coefficient
$T$	transmittance; temperature; kinetic energy in some Hamiltonians	define $T=\mathcal P/\mathcal P_0$ beside the first optical use
$\nu$	ordinary frequency; an occasional label for a vibrational quantum number	use $v$ for a vibrational quantum number when confusion is possible
$\tilde\nu$ or $\sigma$	spectroscopic wavenumber; $\sigma$ also denotes a cross section	prefer $\tilde\nu$ for wavenumber and attach units
$k$	angular wavenumber; Boltzmann constant; rate coefficient	write $k_0=2\pi/\lambda$ and $k_{\mathrm B}$ explicitly
$\sigma$	microscopic cross section; Gaussian standard deviation; wavenumber in some tables	add a subscript such as $\sigma_{\mathrm{abs}}$ or $\sigma_\nu$
$\Gamma$	decay rate; Lorentzian FWHM; Lorentzian HWHM; complex-energy width	state both the observable and its units
$S$	line strength; integrated intensity; action; source function	give the defining integral or matrix element
$I$	irradiance; spectral intensity; detector counts; angular momentum	include the denominator, such as $\mathrm{W\,m^{-2}}$ or counts per bin
$R$	reflectance; resolving power; rotational branch letter	use context-specific subscripts or words

A units column is necessary but not sufficient. Both $\nu$ and a linewidth $\Delta\nu$ can be reported in hertz, and both an absorption cross section and a scattering cross section have units of area.

Spectral Coordinates

Ordinary and angular frequency

The ordinary frequency $\nu$ counts cycles per unit time:

$$[\nu]=\mathrm{Hz}=\mathrm{s^{-1}}.$$

The angular frequency is

$$\omega=2\pi\nu,$$

with units customarily written $\mathrm{rad\,s^{-1}}$. The radian is dimensionless in SI, but retaining “rad” in prose or a table helps prevent a missing factor of $2\pi$.

For a photon,

$$E=h\nu=\hbar\omega.$$

Frequency is fixed by the temporal phase. For a stationary interface, the frequency and angular frequency are continuous across the interface even though wavelength and spatial wavenumber change.

Vacuum and medium wavelength

The vacuum wavelength is

$$\lambda_{\mathrm{vac}} = \frac{c}{\nu}.$$

In a homogeneous medium with phase refractive index $n(\nu)$,

$$\lambda_{\mathrm{med}} = \frac{c}{n(\nu)\nu} = \frac{\lambda_{\mathrm{vac}}}{n(\nu)}.$$

Thus a bare statement such as “the wavelength is $500\,\mathrm{nm}$” is incomplete whenever the vacuum/air distinction is relevant. Atomic line lists may report an observed wavelength or a Ritz wavelength computed from optimized level energies. Those are different provenance labels even when their numerical values agree within uncertainty.

An air wavelength is a medium wavelength inferred using a specified refractive-index formula:

$$\lambda_{\mathrm{air}} = \frac{\lambda_{\mathrm{vac}}}{n_{\mathrm{air}}}.$$

The air conditions and dispersion formula belong in precision metadata. Database defaults can also change with spectral region, so the column header or database documentation is authoritative, not an assumed wavelength range.

Spectroscopic and angular wavenumber

The spectroscopic wavenumber is the reciprocal vacuum wavelength:

$$\tilde\nu = \frac{1}{\lambda_{\mathrm{vac}}} = \frac{\nu}{c}.$$

Its coherent SI unit is $\mathrm{m^{-1}}$, but molecular spectroscopy commonly uses $\mathrm{cm^{-1}}$. The conversion is

$$1\ \mathrm{cm^{-1}} = 100\ \mathrm{m^{-1}}.$$

The photon-energy relation becomes

$$E = hc\tilde\nu.$$

The angular wavenumber, often written $k$, is instead

$$k_{\mathrm{vac}} = \frac{2\pi}{\lambda_{\mathrm{vac}}} = \frac{\omega}{c} = 2\pi\tilde\nu.$$

Therefore “wavenumber” can conceal a factor of $2\pi$. In routine optical spectroscopy, a value in $\mathrm{cm^{-1}}$ almost always denotes $\tilde\nu$, not $k$.

One photon, several coordinates

For a vacuum photon, the basic conversion chain is

$$\begin{aligned} \nu &= \frac{c}{\lambda_{\mathrm{vac}}} = c\tilde\nu, \\ \omega &= 2\pi\nu, \\ k_{\mathrm{vac}} &= 2\pi\tilde\nu, \\ E &= h\nu = \hbar\omega = \frac{hc}{\lambda_{\mathrm{vac}}} = hc\tilde\nu. \end{aligned}$$

Coordinate	Definition	Typical unit	Increases with photon energy?
vacuum wavelength $\lambda_{\mathrm{vac}}$	$c/\nu$	nm, $\mu\mathrm m$, or Å	no
frequency $\nu$	cycles per time	Hz, GHz, THz	yes
angular frequency $\omega$	$2\pi\nu$	$\mathrm{rad\,s^{-1}}$	yes
spectroscopic wavenumber $\tilde\nu$	$1/\lambda_{\mathrm{vac}}$	$\mathrm{cm^{-1}}$	yes
angular wavenumber $k$	$2\pi/\lambda$	$\mathrm{m^{-1}}$	yes
photon energy $E$	$h\nu$	J or eV	yes

Because wavelength runs opposite to energy, a plot that increases from left to right in wavelength decreases from left to right in frequency, wavenumber, and photon energy.

Spectral densities and Jacobians

Changing the horizontal coordinate of a spectrum requires more than relabeling tick marks when the vertical quantity is a density per unit coordinate. If a conserved spectral amount is written

$$dQ = q_x(x)\,dx = q_y(y)\,|dy|,$$

then

$$q_y(y) = q_x(x(y)) \left| \frac{dx}{dy} \right|.$$

For frequency and vacuum wavelength,

$$\left| \frac{d\nu}{d\lambda_{\mathrm{vac}}} \right| = \frac{c}{\lambda_{\mathrm{vac}}^2}.$$

Consequently,

$$q_\lambda(\lambda_{\mathrm{vac}}) = q_\nu(c/\lambda_{\mathrm{vac}}) \frac{c}{\lambda_{\mathrm{vac}}^2}.$$

The peak of a broad density can move under this transformation. A spectrum per nanometre and one per terahertz are not expected to have identical vertical shapes.

For spectroscopic wavenumber expressed in $\mathrm{m^{-1}}$,

$$d\nu = c\,d\tilde\nu.$$

If the numerical wavenumber is instead in $\mathrm{cm^{-1}}$, the Jacobian contains an additional factor of $100$. A normalized line profile must name its coordinate:

$$\int \phi_\nu(\nu)\,d\nu = \int \phi_{\tilde\nu}(\tilde\nu)\,d\tilde\nu = 1.$$

Pointwise transmittance at a physical photon energy is unchanged by reparameterizing the axis. A density, normalized profile, or integrated area does require the Jacobian.

Radiometric Quantities

Let $\mathcal P_0$ be incident radiant power in a declared beam and $\mathcal P$ the power reaching the declared transmitted channel. Detector counts may substitute for power only after establishing proportionality and correcting dark counts, nonlinearity, and wavelength-dependent response.

Transmittance

The transmittance is

$$T = \frac{\mathcal P}{\mathcal P_0}.$$

It is dimensionless. For a passive sample and a consistently defined channel, $0\leq T\leq1$. A value above unity can indicate gain, emission into the collection channel, normalization drift, or a data-reduction error.

Internal transmittance isolates propagation through the sample. Total transmittance can also include interface reflection, scattering, aperture loss, and other losses in the measurement geometry. The two should not be interchanged in a Beer–Lambert analysis.

Percent transmittance is

$$T_{\%}=100T.$$

It is not a new physical quantity; it is a rescaling of the dimensionless ratio.

Decadic and Napierian absorbance

The decadic absorbance is

$$A_{10} = -\log_{10}T = \log_{10}\!\left( \frac{\mathcal P_0}{\mathcal P} \right).$$

The Napierian absorbance is

$$A_{\mathrm e} = -\ln T.$$

They are related by

$$A_{\mathrm e} = (\ln 10)A_{10},$$

and inverted by

$$\begin{aligned} T &= 10^{-A_{10}}, \\ T &= e^{-A_{\mathrm e}}. \end{aligned}$$

Both absorbances are dimensionless. A logarithm does not carry a unit, and a reported absorbance should identify its base. In much of analytical spectroscopy, unqualified absorbance means the decadic quantity; in radiative transfer and AMO theory, the natural-log quantity is more common.

Absorptance is not absorbance

The absorptance is the fraction of incident radiant power actually absorbed. For a passive sample,

$$\mathcal A = \frac{\mathcal P_{\mathrm{absorbed}}}{\mathcal P_0}.$$

It is a linear fraction, not a logarithm. If reflectance $R$ and all other loss channels are negligible,

$$\mathcal A = 1-T.$$

More generally, energy conservation can require

$$1 = T+R+\mathcal A+\mathcal L,$$

where $\mathcal L$ collects scattering or other channels not included in the declared transmitted and reflected beams. Therefore $1-T$ is not automatically the absorptance.

Beer–Lambert Attenuation

Local differential form

For a monochromatic collimated beam propagating through a linear, homogeneous, nonscattering medium with no source term,

$$\frac{dI}{ds} = -\alpha I.$$

Here $I$ is irradiance and $\alpha$ is a Napierian linear attenuation or absorption coefficient, according to which loss mechanisms are included. For constant $\alpha$ over path length $L$,

$$I(L) = I(0)e^{-\alpha L}.$$

Thus

$$T=e^{-\alpha L}.$$

The assumptions matter. Saturation, spatial inhomogeneity, multiple scattering, fluorescence into the detector, interference, or a distributed source term can invalidate this one-parameter interpretation.

Optical depth

The optical depth along a path is

$$\tau = \int_{\mathrm{path}} \alpha(s)\,ds.$$

For the direct beam under the conditions above,

$$T=e^{-\tau}.$$

Therefore

$$\tau = -\ln T = A_{\mathrm e} = (\ln 10)A_{10}.$$

The equality between optical depth and Napierian absorbance is conditional on matching beam and loss definitions. Optical depth may describe total extinction, whereas an experimental absorbance may contain interface and instrument contributions.

Independent layers add in optical depth:

$$\tau_{\mathrm{tot}} = \sum_i\tau_i,$$

while their direct-beam transmittances multiply:

$$T_{\mathrm{tot}} = \prod_i T_i.$$

This additivity is one reason optical depth is useful for heterogeneous paths.

Cross Sections and Macroscopic Coefficients

Microscopic definition

A cross section is an effective area that converts incident particle or photon flux into an event rate. For one target in a weak incident beam,

$$R_{\mathrm{event}} = \sigma\Phi,$$

where $\Phi$ is the incident number flux in $\mathrm{m^{-2}\,s^{-1}}$. Hence

$$[\sigma]=\mathrm{m^2}.$$

An absorption cross section, scattering cross section, photoionization cross section, and extinction cross section answer different event questions. For a passive isolated target,

$$\sigma_{\mathrm{ext}} = \sigma_{\mathrm{abs}} + \sigma_{\mathrm{sca}}.$$

A gain medium can instead be described by a signed net cross section, but the sign convention and population dependence must then be stated.

Number density and column density

For independent absorbers of number density $n(s)$,

$$\alpha_{\mathrm{abs}}(s) = n(s)\sigma_{\mathrm{abs}}.$$

If the cross section is constant along the path,

$$\tau_{\mathrm{abs}} = \sigma_{\mathrm{abs}}N_{\mathrm{col}},$$

where the column density is

$$N_{\mathrm{col}} = \int n(s)\,ds.$$

The dimensions close:

$$[N_{\mathrm{col}}] = \mathrm{m^{-2}}, \qquad [\sigma N_{\mathrm{col}}] = 1.$$

When multiple species or internal states contribute,

$$\alpha(\nu,s) = \sum_j n_j(s)\sigma_j(\nu).$$

The state populations are part of the model. A database line strength does not by itself specify the cross section of an unequilibrated sample.

Molar absorption coefficients

The common decadic Beer–Lambert form is

$$A_{10} = \varepsilon_{10}c_{\mathrm M}L,$$

where $c_{\mathrm M}$ is molar concentration and the units of $\varepsilon_{10}$ must match those chosen for concentration and path length. A frequently used combination is

$$[\varepsilon_{10}] = \mathrm{L\,mol^{-1}\,cm^{-1}}, \quad [c_{\mathrm M}] = \mathrm{mol\,L^{-1}}, \quad [L] = \mathrm{cm}.$$

For those units, the corresponding microscopic cross section in $\mathrm{m^2}$ is

$$\sigma_{\mathrm{abs}} = \frac{\ln 10}{10N_{\mathrm A}} \varepsilon_{10}.$$

The factor $10$ is a unit-conversion factor, not spectroscopy. It combines litres with cubic metres and centimetres with metres. In coherent SI units, one can instead write

$$\alpha_{\mathrm e} = \kappa_{\mathrm e}c,$$

with $\kappa_{\mathrm e}$ in $\mathrm{m^2\,mol^{-1}}$ and $c$ in $\mathrm{mol\,m^{-3}}$, giving

$$\sigma_{\mathrm{abs}} = \frac{\kappa_{\mathrm e}}{N_{\mathrm A}}.$$

Line-centre and integrated cross sections

A line-centre cross section is a pointwise area:

$$\sigma_0 = \sigma(\nu_0).$$

An integrated cross section includes the chosen spectral coordinate:

$$\Sigma_\nu = \int \sigma(\nu)\,d\nu.$$

Its units are $\mathrm{m^2\,Hz}$, not $\mathrm{m^2}$. An integral over wavenumber or angular frequency has a different numerical value and different units. If

$$\sigma(\nu) = \Sigma_\nu\phi_\nu(\nu),$$

then

$$\int\phi_\nu(\nu)\,d\nu=1,$$

so $\phi_\nu$ has units $\mathrm{Hz^{-1}}$. Never compare a line-centre cross section with an integrated cross section, or two integrated values on different coordinates, without converting the measure.

The Oscillator Strength Reference owns the corresponding integrated-strength relations.

Attenuation Vocabulary

The following table separates quantities that are often all called “absorption.”

Quantity	Definition	Units	Main warning
transmittance $T$	transmitted/incident radiant power	1	depends on collection channel
decadic absorbance $A_{10}$	$-\log_{10}T$	1	state the logarithm base
Napierian absorbance $A_{\mathrm e}$	$-\ln T$	1	equals $\tau$ only for matched definitions
absorptance $\mathcal A$	absorbed/incident radiant power	1	not $A_{10}$ and not always $1-T$
optical depth $\tau$	$\int\alpha\,ds$	1	identify absorption or total extinction
linear coefficient $\alpha$	local fractional loss per length	$\mathrm{m^{-1}}$	identify included loss processes
cross section $\sigma$	event rate per target divided by number flux	$\mathrm{m^2}$	name the event and internal state
column density $N_{\mathrm{col}}$	$\int n\,ds$	$\mathrm{m^{-2}}$	state species and state population
molar decadic coefficient $\varepsilon_{10}$	$A_{10}/(c_{\mathrm M}L)$	often $\mathrm{L\,mol^{-1}\,cm^{-1}}$	units are convention dependent

The older phrase extinction coefficient is especially ambiguous: it has been used for a Napierian attenuation coefficient, a decadic coefficient, a molar absorptivity, and the imaginary part of a refractive index. Prefer the specific modern quantity name and equation.

Linewidth Nomenclature

A linewidth needs four labels

A reproducible linewidth should specify:

1. the profile or operational estimator;
2. full width, half width, standard deviation, or decay-rate convention;
3. the spectral coordinate and unit; and
4. whether the width is observed, instrument-corrected, homogeneous, inhomogeneous, or model decomposed.

For example, “Gaussian FWHM $\Delta\nu_{\mathrm G}=12.0\,\mathrm{MHz}$ after instrument-response deconvolution” is interpretable. “$\Gamma=12$” is not.

Common width parameters

For a Gaussian profile with standard deviation $\sigma_\nu$,

$$\Delta\nu_{\mathrm{FWHM}}^{(\mathrm G)} = 2\sqrt{2\ln2}\,\sigma_\nu.$$

For a Lorentzian written with HWHM $\gamma_\nu$,

$$L(\nu) \propto \frac{\gamma_\nu} {(\nu-\nu_0)^2+\gamma_\nu^2},$$

the full width is

$$\Delta\nu_{\mathrm{FWHM}}^{(\mathrm L)} = 2\gamma_\nu.$$

A symbol $\Gamma$ may denote either of these widths or an angular-frequency decay rate. The equation defining $\Gamma$ is therefore part of the result.

Label	Meaning	Conversion
Gaussian $\sigma_\nu$	standard deviation	FWHM $=2\sqrt{2\ln2}\,\sigma_\nu$
Lorentzian $\gamma_\nu$	HWHM	FWHM $=2\gamma_\nu$
FWHM	separation of the two half-maximum points	profile dependent
HWHM	half of FWHM for a symmetric profile	FWHM $=2\,\mathrm{HWHM}$
$1/e$ half-width	offset where amplitude or intensity falls to $1/e$	state amplitude versus intensity
decay rate	inverse time constant	relation to spectral width depends on coherence convention

The Line Shape Reference gives the canonical normalized profiles, convolution rules, and lifetime-broadening relations.

Narrow-line coordinate conversion

For a line narrow enough that the coordinate transformation is nearly linear across its width,

$$\Delta\omega \simeq 2\pi\Delta\nu,$$ $$\Delta\tilde\nu \simeq \frac{\Delta\nu}{c},$$

where $\tilde\nu$ is in $\mathrm{m^{-1}}$, and

$$\Delta\lambda_{\mathrm{vac}} \simeq \frac{\lambda_{\mathrm{vac}}^2}{c} \Delta\nu.$$

The last equation gives a positive width; the derivative $d\lambda/d\nu$ itself is negative. For broad or asymmetric features, transform the two endpoints or the full profile rather than applying a single central Jacobian.

Resolving power is a dimensionless instrument or observation metric,

$$\mathcal R = \frac{x_0}{\Delta x},$$

for a declared coordinate and width convention. It is not itself a physical linewidth.

Molecular Branch Notation

Upper and lower state labels

Molecular spectroscopy commonly marks upper-state quantum numbers with a prime and lower-state quantum numbers with a double prime:

$$J' \quad\text{and}\quad J''.$$

For absorption, define

$$\Delta J = J'-J''.$$

The standard rotational branch letters are:

Branch	$\Delta J$	Simple absorption assignment
O	$-2$	$J'=J''-2$
P	$-1$	$J'=J''-1$
Q	$0$	$J'=J''$
R	$+1$	$J'=J''+1$
S	$+2$	$J'=J''+2$

Which branches are allowed depends on transition type, molecular symmetry, and angular-momentum coupling. The existence of a letter in this table is not a selection rule.

Parenthesized lower-state quantum number

In the common simple-diatomic convention, the integer in a label such as $R(3)$ is the lower-state rotational quantum number:

$$\begin{aligned} R(3):\quad &J''=3\longrightarrow J'=4, \\ P(3):\quad &J''=3\longrightarrow J'=2, \\ Q(3):\quad &J''=3\longrightarrow J'=3. \end{aligned}$$

Emission papers and specialized databases may order or label states differently. A line list should define its convention rather than relying on the reader to infer the direction.

A vibronic band is often identified by

$$(v',v''),$$

with upper and lower vibrational quantum numbers in that order. Thus the $(1,0)$ band and the line $R(3)$ within it encode different parts of the assignment.

Multiple angular momenta

Open-shell molecules can carry both rotational angular momentum excluding spin, often $N$, and total angular momentum including spin, $J$. A compound branch label can then encode both

$$\Delta N=N'-N'' \quad\text{and}\quad \Delta J=J'-J''.$$

For example, a source may use a two-letter label in which one letter tracks $\Delta N$ and the other tracks $\Delta J$. Superscripts and subscripts may also identify spin components, parity, $K$, or fine-structure ladders. There is no safe universal decoding rule without the source’s definition.

The minimum defensible practice is to print the relevant upper and lower quantum numbers in a table and treat the compact branch string as a convenience label. See Rotational Spectroscopy and Term Symbol Reference for the underlying angular-momentum labels.

Reporting Checklist

A spectrum or fitted line table is reproducible when it records:

• the horizontal coordinate, unit, and vacuum/air/medium convention;
• the refractive-index equation and environmental conditions for air or medium wavelengths;
• observed, calculated, or Ritz provenance for line positions;
• the vertical observable and whether it is a density per unit coordinate;
• the incident and transmitted channels used to define $T$;
• logarithm base for absorbance;
• absorption, scattering, or total-extinction scope for $\alpha$, $\tau$, and $\sigma$;
• species, isotope, internal state, temperature, pressure, and column density where relevant;
• line-profile model, width convention, coordinate, and instrument-response treatment;
• upper and lower quantum-number definitions for branch labels;
• calibration, baseline, and uncertainty procedures; and
• database release, query date, and original data source.

For machine-readable work, store the convention in metadata rather than in a plot caption alone. A numerical column named width or intensity is not a stable data model.

Common Mistakes

Treating air wavelength as photon energy divided by a different constant

Photon energy is determined by frequency:

$$E=h\nu.$$

At a stationary interface $\nu$ does not change. The medium wavelength changes because the phase velocity changes; the photon has not acquired a different vacuum frequency merely because $\lambda_{\mathrm{med}}$ is shorter.

Confusing reciprocal centimetres with angular wavenumber

A spectroscopic value in $\mathrm{cm^{-1}}$ normally means $\tilde\nu=1/\lambda_{\mathrm{vac}}$. Inserting it as $k=2\pi/\lambda$ introduces a factor of $2\pi$.

Converting a density without its Jacobian

Replacing $\nu$ by $c/\lambda$ on the horizontal axis while leaving a “per hertz” vertical density unchanged does not conserve integrated area. Transform both coordinates and density.

Calling one minus transmittance the absorbance

$1-T$ is a linear missing-power fraction. Absorbance is logarithmic:

$$A_{10}=-\log_{10}T.$$

Even the missing-power fraction is not necessarily absorptance when reflection and scattering occur.

Omitting the logarithm base

For the same transmittance,

$$A_{\mathrm e} = (\ln10)A_{10}.$$

A fitted concentration or cross section will inherit the factor-of-$\ln10$ error if the base is guessed incorrectly.

Mixing line-centre and integrated cross sections

$\sigma(\nu_0)$ has units of area. $\int\sigma(\nu)d\nu$ has units of area times frequency. Their ratio depends on linewidth and line shape.

Reporting a bare linewidth

Neither “$10\,\mathrm{MHz}$” nor “$\Gamma=10\,\mathrm{MHz}$” reveals whether the value is FWHM, HWHM, a Gaussian standard deviation, an angular rate divided by $2\pi$, or an instrument-limited width.

Reading a branch letter as a complete assignment

$R$ says $\Delta J=+1$ only in the declared branch convention. It does not specify electronic state, vibrational band, parity, spin component, or whether another angular momentum also changes.

Exercises

Exercise 1: Convert a vacuum wavelength

A transition has vacuum wavelength $\lambda_{\mathrm{vac}}=780.000\,\mathrm{nm}$. Find $\nu$, $\omega$, $\tilde\nu$ in $\mathrm{cm^{-1}}$, and the photon energy in eV.

Solution

Using the exact SI values of $c$, $h$, and the elementary charge,

$$\nu = \frac{c}{\lambda_{\mathrm{vac}}} = 3.84349305\times10^{14}\ \mathrm{Hz}.$$

Then

$$\omega = 2\pi\nu = 2.41493791\times10^{15}\ \mathrm{rad\,s^{-1}}.$$

The spectroscopic wavenumber is

$$\tilde\nu = \frac{1}{780.000\times10^{-7}\ \mathrm{cm}} = 1.28205128\times10^4\ \mathrm{cm^{-1}}.$$

Finally,

$$\frac{E}{e} = \frac{hc}{e\lambda_{\mathrm{vac}}} = 1.58954101\ \mathrm{eV}.$$

The wavelength carries six significant digits, so reporting more than six significant digits in the results would not represent measurement precision.

Exercise 2: Vacuum and air wavelength

A line has $\lambda_{\mathrm{vac}}=500.000\,\mathrm{nm}$. Using $n_{\mathrm{air}}=1.000277$, find its air wavelength. What happens to its frequency at a stationary vacuum–air interface?

Solution

The air wavelength is

$$\lambda_{\mathrm{air}} = \frac{500.000\ \mathrm{nm}}{1.000277} = 499.8615\ \mathrm{nm}.$$

The frequency remains

$$\nu = \frac{c}{\lambda_{\mathrm{vac}}} = 5.99584916\times10^{14}\ \mathrm{Hz}.$$

The shorter medium wavelength reflects the smaller phase velocity, not a change in frequency.

Exercise 3: Three attenuation coordinates

A sample has transmittance $T=0.200$. Find the decadic absorbance, Napierian absorbance, and optical depth for a pure-attenuation direct-beam model.

Solution

The decadic absorbance is

$$A_{10} = -\log_{10}(0.200) = 0.698970.$$

The Napierian absorbance is

$$A_{\mathrm e} = -\ln(0.200) = 1.609438.$$

Under the stated model,

$$\tau = A_{\mathrm e} = 1.609438.$$

The numerical values differ because absorbance conventions use different logarithm bases.

Exercise 4: Column density and cross section

A uniform ensemble has column density $N_{\mathrm{col}}=2.00\times10^{20}\,\mathrm{m^{-2}}$ and absorption cross section $\sigma=3.00\times10^{-21}\,\mathrm{m^2}$. Find $\tau$, $T$, and $A_{10}$.

Solution

The optical depth is

$$\tau = N_{\mathrm{col}}\sigma = 0.600.$$

Hence

$$T = e^{-0.600} = 0.548812,$$

and

$$A_{10} = \frac{\tau}{\ln10} = 0.260577.$$

Multiplying $T$ by $100$ gives $54.8812\%$ transmittance; it does not give the absorbance.

Exercise 5: Molar coefficient to microscopic cross section

A solution has $\varepsilon_{10}=1.50\times10^4\, \mathrm{L\,mol^{-1}\,cm^{-1}}$, concentration $c_{\mathrm M}=20.0\,\mu\mathrm{mol\,L^{-1}}$, and path length $L=1.00\,\mathrm{cm}$. Find $A_{10}$, $T$, $\tau$, and the corresponding microscopic cross section.

Solution

The concentration is

$$c_{\mathrm M} = 2.00\times10^{-5}\ \mathrm{mol\,L^{-1}}.$$

Therefore

$$A_{10} = \varepsilon_{10}c_{\mathrm M}L = 0.300.$$

The transmittance and optical depth are

$$\begin{aligned} T &= 10^{-0.300} = 0.501187, \\ \tau &= (\ln10)(0.300) = 0.690776. \end{aligned}$$

Using the stated litre–mole–centimetre convention,

$$\begin{aligned} \sigma &= \frac{\ln10}{10N_{\mathrm A}} \varepsilon_{10} \\ &= 5.735\times10^{-21}\ \mathrm{m^2}. \end{aligned}$$

The conversion would change if the numerical value of $\varepsilon$ used a different concentration or length unit.

Exercise 6: Convert a narrow linewidth

A line is centred at $\nu_0=500.0\,\mathrm{THz}$ and has ordinary-frequency FWHM $\Delta\nu=100.0\,\mathrm{MHz}$. Find the central vacuum wavelength and the approximate FWHM in angular frequency, $\mathrm{cm^{-1}}$, and picometres.

Solution

The central wavelength is

$$\lambda_0 = \frac{c}{\nu_0} = 599.5849\ \mathrm{nm}.$$

The angular-frequency width is

$$\Delta\omega = 2\pi\Delta\nu = 6.28319\times10^8\ \mathrm{rad\,s^{-1}}.$$

The spectroscopic-wavenumber width is

$$\begin{aligned} \Delta\tilde\nu &= \frac{\Delta\nu}{c} \\ &= 0.00333564\ \mathrm{cm^{-1}}. \end{aligned}$$

The wavelength width is

$$\begin{aligned} \Delta\lambda &\simeq \frac{\lambda_0^2}{c}\Delta\nu \\ &= 0.119917\ \mathrm{pm}. \end{aligned}$$

Because $\Delta\nu/\nu_0=2.0\times10^{-7}$, the local linear conversion is well justified.

Exercise 7: Decode branch labels

In a simple diatomic absorption spectrum, decode $R(4)$, $P(4)$, and $Q(4)$ in terms of $J''$ and $J'$. Does the notation alone prove all three lines are allowed?

Solution

The parenthesized number labels the lower state:

$$\begin{aligned} R(4):\quad &J''=4,\quad J'=5, \\ P(4):\quad &J''=4,\quad J'=3, \\ Q(4):\quad &J''=4,\quad J'=4. \end{aligned}$$

The labels encode $\Delta J=+1,-1,0$, respectively. They do not prove that the branches are allowed. Electronic symmetry, transition multipolarity, parity, spin, and other angular momenta determine the actual selection rules.

Exercise 8: Audit an ambiguous line record

A data file contains the row A = 0.50, sigma = 12500 cm^-1, Gamma = 10 MHz, S = 2.1e-20. List the minimum questions needed before using it in a model.

Solution

At least the following information is missing:

• whether $A$ is decadic or Napierian absorbance, an Einstein coefficient, or another quantity;
• if it is absorbance, the incident and transmitted channels and logarithm base;
• whether “sigma” is the spectroscopic vacuum wavenumber $\tilde\nu=1/\lambda_{\mathrm{vac}}$ or another symbol;
• whether the position is observed or Ritz/calculated;
• whether $\Gamma$ is FWHM, HWHM, Gaussian standard deviation, or a decay rate expressed after division by $2\pi$;
• whether the width is observed, deconvolved, homogeneous, or instrument-limited;
• what $S$ means, its units, and the coordinate used by any defining integral;
• which species, isotope, states, temperature, and pressure the row describes; and
• the source, uncertainty, and database release.

The row cannot be repaired by guessing from the numerical magnitudes. Its schema needs physical definitions.

Cross-Links

• Spectroscopy
• Constants and Conversions
• Line Shape Reference
• Oscillator Strength Reference
• Einstein Coefficient Reference
• Absorption and Emission
• Line Shapes and Broadening
• Rotational Spectroscopy
• Vibrational Spectroscopy
• Infrared Spectroscopy
• Electronic Spectroscopy
• Precision Spectroscopy
• Term Symbol Reference
• Selection Rule Tables

References

• A. Kramida, “Atomic Spectroscopy: An Introduction”, National Institute of Standards and Technology, including the frequency–wavenumber–vacuum-wavelength convention, accessed 2026-07-26.
• A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, “Help for Spectral Lines”, NIST Atomic Spectra Database, including observed and Ritz values and vacuum/air wavelength conventions, accessed 2026-07-26.
• International Union of Pure and Applied Chemistry, “Transmittance”, “Absorbance”, “Decadic absorbance”, and “Napierian absorbance”, Compendium of Chemical Terminology, 5th ed., online version 5.0.0, 2025.
• International Union of Pure and Applied Chemistry, “Absorptance”, “Absorption coefficient”, “Linear Napierian absorption coefficient”, “Attenuation coefficient”, and “Net absorption cross section”, Compendium of Chemical Terminology, 5th ed., online version 5.0.0, 2025.
• International Union of Pure and Applied Chemistry, “Beer–Lambert law”, Compendium of Chemical Terminology, 5th ed., online version 5.0.0, 2025.
• H. Goenaga-Infante et al., “Glossary of methods and terms used in analytical spectroscopy,” Pure and Applied Chemistry 93, 647–776 (2021), doi:10.1515/pac-2019-0203.
• C. J. H. Schutte, J. E. Bertie, P. R. Bunker, J. T. Hougen, I. M. Mills, J. K. G. Watson, and B. P. Winnewisser, “Notations and conventions in molecular spectroscopy: Part 1. General spectroscopic notation,” Pure and Applied Chemistry 69, 1633–1640 (1997), doi:10.1351/pac199769081633.
• HITRAN, Definitions and Units, including line intensity, absorption coefficient, and optical-depth conventions, accessed 2026-07-26.
• R. C. Hilborn, “Einstein coefficients, cross sections, $f$ values, dipole moments, and all that,” American Journal of Physics 50, 982–986 (1982), doi:10.1119/1.12937; revised version.
• P. F. Bernath, Spectra of Atoms and Molecules, 5th ed., Oxford University Press, 2025, doi:10.1093/oso/9780197754498.001.0001.
• J. M. Hollas, Modern Spectroscopy, 4th ed., Wiley, 2004, doi:10.1002/0470094719.