How it is calculated
Level energies \(E\) are term values in cm⁻¹ above the ground level, exactly as NIST ASD lists them. A scheme starts from a ground-term component \(E_g\) (0 for the lowest), reaches the resonant level \(E_r\) with \(n\) identical photons, and ionizes with \(m\) photons of the same color (or of a second color \(\lambda_2\)). The resonance condition and the vacuum wavelength are
The air wavelength is \(\lambda_\text{air} = \lambda_\text{vac}/n_\text{air}(\lambda)\) with the dispersion of standard air (15 °C, 101.325 kPa, dry). The default is the five-parameter formula of Peck and Reeder that NIST ASD itself uses, with \(\sigma = 1/\lambda\) in µm⁻¹,
valid from 185 nm to 1.7 µm; Ciddor’s 1996 two-term formula (450 ppm CO₂) is offered as an alternative. At 243 nm the index is 1.0003035, so the air wavelength is 0.074 nm shorter than the vacuum value: a dye laser’s wavemeter usually reads vacuum, a grating’s calibration often air, and that difference is many Doppler widths.
The ionizing step takes the species to \(E_\text{after} = E_r + m\,h\nu_2\) (with \(h\nu_2 = h\nu\) for one color). It ionizes if \(E_\text{after} \ge \text{IE}\), and the surplus is the photoelectron kinetic energy when the ion is left in its ground state:
If the ion can be left in an excited spin-orbit or vibrational level, a second, slower photoelectron group appears, lower by that excitation (Xe⁺ ²P°₁/₂ lies 10,537 cm⁻¹ above ²P°₃/₂; the calculator reports only the ground-state channel). The heavy ion itself takes a negligible share of the energy.
A particle moving with speed \(v\) along the beam sees each photon shifted by \(\nu\,v/c\), so an \(n\)-photon resonance driven by co-propagating photons shifts by
twice the one-photon shift at the same wavelength for \(n = 2\). With one photon from each of two counter-propagating beams the first-order shifts cancel (Doppler-free two-photon spectroscopy); the calculator reports the co-propagating value, which is also the shift of a 1+1 scheme and the full width you must tune across to see every velocity in a beam. For the Doppler width of a thermal sample, use the line-broadening calculator.
Fine-structure companions are computed from the same upper level and every component of the ground term, and flagged when |ΔJ| exceeds \(n\). The laser suggestions test the target wavelength against fixed Nd:YAG harmonics (\(1064.1/k\) nm), each dye’s tuning range directly, doubled (\(2\lambda\)), mixed with the 1064.1 nm fundamental (\(1/\lambda_\text{dye} = 1/\lambda - 1/1064.1\)), doubled and then mixed with 1064.1 nm, or tripled; and the Ti:sapphire and OPO ranges with their harmonics.
How REMPI works and why it is used
A single photon ionizes an atom only in the vacuum ultraviolet (13.6 eV = 91 nm for hydrogen), where lasers are awkward and windows are opaque. Multiphoton ionization with a focused pulsed laser avoids that, but the non-resonant rate scales as \(I^{n+m}\) and is tiny. Placing a real level at the energy of the first \(n\) photons changes everything: the first step becomes a two-photon transition whose line-integrated cross section is a few 10⁻³⁵ cm⁴ ((2.66 ± 0.80) × 10⁻³⁵ cm⁴ for O 3p ³P ← 2p⁴ ³P₂, Bamford, Jusinski and Bischel; 2–7 × 10⁻³⁵ cm⁴ for the Xe 6p levels, Kröll and Bischel), which spread over a Doppler width of several GHz is of order 10⁻⁴⁵ cm⁴ s at line centre. One more photon then ionizes the excited atom with a cross section of 10⁻¹⁹–10⁻¹⁸ cm² (5.3 × 10⁻¹⁹ cm² for O 3p ³P at 226 nm; 4.3 × 10⁻¹⁸ cm² for Xe 6p[1/2]₀ at 248 nm), large enough to saturate with a few mJ focused to a fraction of a millimetre. The result is that a large share of the atoms in the focal volume that sit in the probed quantum state end up as ions (Kröll and Bischel estimate 10 % of the xenon atoms at 30 MW cm⁻² for 25 ns), which a channel plate detects with near-unit efficiency.
The ions are usually mass-analysed in a time-of-flight spectrometer, which separates the probed species from everything else the laser ionizes, or imaged in a velocity-map imaging (VMI) apparatus, where the ion (or photoelectron) velocity distribution appears directly as a ring pattern. Because the signal tracks one quantum state, scanning the laser over the rotational or fine-structure components gives a state distribution; that is how photodissociation product branching ratios (I/I*, O(³P)/O(¹D), Cl/Cl*) and nascent rotational distributions are measured. The Thermochemistry Lab covers the energetics that decide which products appear. One caution for molecules: the ion keeps absorbing during the same pulse. Boesl, Neusser and Schlag showed with time-delayed two-color pulses on benzene that fragmentation proceeds by climbing two independent ladders, one in the neutral up to the ionization limit and a second in the parent ion (ladder switching), so fragment ions grow with pulse energy at the parent’s expense.
Why 2+1 for atoms: the first excited levels of light atoms lie at 60,000–100,000 cm⁻¹ (O 3p at 88,600, N 3p at 94,900, H 2s at 82,259 cm⁻¹). Two photons of 200–250 nm, which one doubled dye laser or a Ti:sapphire harmonic produces, reach them, and the third photon of the same beam comfortably clears the limit because the intermediate level sits above half the ionization energy. Two-photon excitation also connects levels of the same parity, which is exactly where the np levels sit relative to the np⁴ or np⁵ ground states. Why 1+1 for molecules: NO, benzene, aromatics and many radicals have bound excited states at 30,000–45,000 cm⁻¹, above half their (lower) ionization energy, so one UV photon reaches the state and a second ionizes; the one-photon step has a large cross section, so 1+1 saturates easily and even runs with unfocused beams. Molecules whose first bright states are too low for 1+1, or whose Rydberg states lie at 80,000–100,000 cm⁻¹ (CO, H₂, H₂O, HCl, OH), are done 2+1.
Selection rules
One photon changes parity and allows ΔJ = 0, ±1 (not 0 ↔ 0), ΔL = ±1 for a one-electron jump, and ΔS = 0 in LS coupling. Two photons absorbed from a single beam act as the product of two dipole operators, a tensor of rank 0 (scalar) plus rank 2: the parity does not change, |ΔJ| ≤ 2 with J = 0 ↔ 1 forbidden and J = 0 ↔ 0 allowed only through the scalar part (it vanishes for two photons of the same circular polarization from one beam, which is sometimes used to clean up a spectrum, and survives with linear or with opposite circular polarizations), Δl = 0, ±2 for the active electron (s → s, s → d, p → p, p → f), and ΔS = 0. Bonin and McIlrath tabulate the complete polarization-dependent rules (same parity, |ΔF| ≤ 2, the forbidden 0 ↔ 1 and the polarization-dependent 0 ↔ 0 cases), including those that reopen when the two photons differ in frequency or polarization; the ΔS = 0 rule and the three-photon rules are the standard angular-momentum results found in Demtröder. Three photons behave like one as far as parity goes (odd), with |ΔJ| ≤ 3. The calculator applies the parity, ΔJ, and (for LS terms) ΔS rules and marks the J = 0 ↔ 1 exception; it does not estimate line strengths.
For molecules the same ideas hold with the vibronic symmetry in place of parity: for centrosymmetric molecules two-photon transitions are g ↔ g (the H₂ E,F ¹Σg⁺ ← X ¹Σg⁺ and O₂ C ³Πg ← X ³Σg⁻ schemes), one-photon transitions g ↔ u. Rotational branches O, P, Q, R, S (ΔJ = −2 … +2) appear in two-photon spectra, and their relative intensities depend on polarization, as Tjossem and Smyth’s 200:1 linear-to-circular ratio for the CO B–X Q branch shows.
Practical notes
- Saturation of the ionizing step. With σ ≈ 10⁻¹⁸ cm² and a photon fluence of 10¹⁸ cm⁻² (about 60 µJ at 250 nm in a 100 µm spot, or 1 mJ in a 0.4 mm spot; Kröll and Bischel’s 30 MW cm⁻² for 25 ns is 9 × 10¹⁷ photons cm⁻²) the excited atom ionizes with near-certainty; the signal then depends only on the two-photon step, roughly as \(I^2\). Below saturation it goes as \(I^3\). Measuring the power dependence tells you where you are.
- Power broadening and AC Stark shifts. The same intensity that saturates the last step shifts and broadens the two-photon resonance. For the H 1S–2S line at 243 nm the AC Stark coefficient is 1.67 Hz per W cm⁻² (Kolachevsky et al.), i.e. 0.06 cm⁻¹ (1.7 GHz) at 10⁹ W cm⁻² and 0.6 cm⁻¹ at 10¹⁰ W cm⁻², and the same intensities photoionize the 2S atoms within the pulse, which broadens the line further. Take spectra at several pulse energies if the line position matters.
- Doppler width versus laser bandwidth. A room-temperature O atom has a Doppler FWHM of about 0.27 cm⁻¹ (8 GHz) on the 226 nm two-photon line (twice the one-photon value), a cold beam far less; a pulsed dye laser’s doubled bandwidth of 0.1–0.3 cm⁻¹ is comparable, and a transform-limited 5 ns pulse has 0.003 cm⁻¹. Compare them with the line-broadening and pulse-bandwidth calculators; the narrower of the two sets the state selectivity, the wider sets how far you must scan to collect every velocity class (the Doppler-shift result above).
- Photoelectron energy for VMI calibration. Xe 2+1 at 249.63 nm releases 2.770 eV electrons for Xe⁺ ²P°₃/₂ and 1.464 eV for ²P°₁/₂; O 2+1 at 225.65 nm gives 2.865 eV; the atom’s known kinetic energies calibrate the velocity map. Molecules leave the ion vibrationally excited, so their photoelectron spectra carry a progression rather than one ring.
- Ion masses and flight times. One-color REMPI also ionizes the precursor and makes fragment ions; the time-of-flight calculator gives arrival times and the mass resolution needed to separate, say, O⁺ (m/z 16) from CH₄⁺ at the same mass number. Light ions arrive first, so a 2+1 H-atom signal at m/z 1 sits well clear of everything else.
- Non-resonant background. At the intensities needed for 2+1, the laser also ionizes precursor molecules non-resonantly and photolyses them; a one-color probe therefore often creates the very atoms it detects. Compare on- and off-resonance signals, vary the pulse energy, or use a two-color scheme with a weak resonant beam and a strong ionizing beam.
- Wavelength calibration. The resonance wavelength is the quantity to set, but check which medium your reference uses: ASD lines are listed in air above 200 nm and in vacuum below; wavemeters read vacuum unless told otherwise. The known-scheme table gives both.
Typical values
Every documented scheme on this page, recomputed from the level energies (NIST ASD excerpts for atoms; WebBook ionization energies and band origins for molecules). Wavelengths are for the lowest ground component; the fine-structure companions appear when you select the scheme above.
| Species | Scheme | λvac (nm) | λair (nm) | Resonance (cm⁻¹) | Margin over IE (eV) | Primary source |
|---|
Numbers that the primary paper quotes agree with the recomputed values within the 0.01–0.1 nm precision the papers give (the N 3p ⁴D° components spread over 210.79–211.04 nm and the S 4p ³P ones over 308.2–311.0 nm). Schemes that could not be traced to a paper whose record or abstract was checked (for example the Br* and O 3p ⁵P lines sometimes quoted) are deliberately left out, and so is NH₃ 2+1 via B̃: Stanley, Echt and Castleman report its bands only as a 304–340 nm range and no archived source gives a band origin. See the references for what each source supports.
Worked example
The default scheme, H 2+1 via 2s ²S₁/₂ at 243.13 nm. NIST ASD puts 2s at 82,258.9544 cm⁻¹ and the ionization limit at 109,678.7717 cm⁻¹.
- Each of the two photons carries 82,258.954 / 2 = 41,129.477 cm⁻¹ = 5.0994 eV, so \(\lambda_\text{vac}\) = 10⁷ / 41,129.477 = 243.1346 nm. In standard air, \(n\) = 1.0003035, \(\lambda_\text{air}\) = 243.0609 nm.
- The third photon takes the atom to 3 × 41,129.477 = 123,388.43 cm⁻¹, which is 13,709.66 cm⁻¹ = 1.6998 eV above the limit: the scheme ionizes, and the photoelectron leaves at \(\sqrt{2 \times 1.6998\ \text{eV}/m_e}\) = 773 km/s.
- Selection rules: 1s and 2s are both even parity with J = 1/2 → 1/2, allowed for two photons through the scalar part (linear polarization). Deuterium’s 2s lies 22.38 cm⁻¹ higher, moving the line to 243.0685 nm, 0.066 nm to the blue.
- A hydrogen atom from a photodissociation with 2000 m/s along the beam shifts the two-photon resonance by 82,259 × 2000 / 2.998 × 10⁸ = 0.549 cm⁻¹ = 16.5 GHz (0.0016 nm). Scanning over the Doppler profile of fast fragments therefore needs a few cm⁻¹; counter-propagating beams would remove the shift for the Doppler-free pair.
- Laser: 243.13 nm is the doubled output of a dye at 486.3 nm (Coumarin 102 or Coumarin 307, on a 355 nm or a 308 nm pump) or the third harmonic of a Ti:sapphire at 729.4 nm.
Assumptions and limits
- Energetics only. The calculator says whether a scheme is allowed and whether it ionizes; it does not estimate two-photon cross sections, line strengths, or the actual ion yield, which depend on intensity, bandwidth, focusing, and the particular level. The cited papers give cross sections for O and Xe.
- Level excerpts. Each atom carries only the ground-term components, a handful of low Rydberg levels, and the first ionization limit, copied from NIST ASD 5.12 with the uncertainties ASD lists (several atoms carry none). For every other level follow the ASD link; for higher Rydberg or autoionizing levels (for example O(¹D) detection through 2p³(²D°)3p ¹F₃ above the limit) type the level energy in design mode.
- Molecules. The resonant energy is a band origin (or, for H₂O, the band head) from the cited paper or from Huber–Herzberg constants in the WebBook; rotational structure, vibrational bands other than the one listed, and the ion’s internal energy are not modeled, so the photoelectron energy is the maximum for that band.
- Air wavelengths. Standard dry air at 15 °C and 101.325 kPa. At 20 °C and typical humidity the index is lower by about 5 × 10⁻⁷ (0.0001 nm at 250 nm), below the precision of the schemes. No air value is given below 185 nm, where the formula is undefined and air absorbs anyway.
- Doppler shift. First order, non-relativistic, co-propagating photons; the second-order shift is 10⁻¹⁰ at 2000 m/s.
- Laser ranges are catalog and datasheet values for broadband operation and typical conversion; a given dye laser, crystal set, or OPO reaches somewhat less at the edges, and narrowband operation narrows the range further.