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Experiment 12 — Molecular Fluorescence, the Stokes Shift, and Franck–Condon

Objectives

By the end of this experiment you should be able to:

Textbook connection

Read §12.4–12.7. Chapter 12 builds molecular energy levels as a hierarchy: electronic states separated by a few eV, vibrational levels within them separated by tenths of an eV, and rotational levels by meV. These energy scales and the fact that electronic transitions are fast relative to nuclear motion help explain the spectra measured here.

Theory

Why absorption and emission are not at the same wavelength

The Franck–Condon principle approximates an electronic transition as fast compared with nuclear motion. On a potential-energy diagram, the most probable transitions are therefore vertical.

The excited electronic state may have a different equilibrium nuclear geometry from the ground state. A vertical transition from a populated ground-state vibrational level can therefore populate several excited-state levels. Vibrational and solvent relaxation often precede fluorescence from the lowest excited singlet state. The vertical emission transition can then populate vibrationally excited ground-state levels, which relax afterward.

Ground and excited electronic potential energy curves offset in bond length, with a vertical absorption transition, diagonal vibrational relaxation, a vertical emission transition at lower energy, and a second relaxation back to the ground vibrational level, plus a comparison of the absorption and emission photon energies showing the Stokes shift.

Figure 12.1:An idealized Franck–Condon cycle. Vertical arrows show electronic transitions; dashed arrows show relaxation between them. The pictured excitation and emission photons differ in energy. Real bands also depend on solvent reorganization, populated starting levels, and other decay channels.

For the usual relaxed fluorescence from these dilute dyes, the emission-band maximum is expected at lower photon energy than the corresponding absorption-band maximum. Their difference is the Stokes shift,

Δν~Stokes=ν~abs,max−ν~em,max,\Delta\tilde\nu_{\text{Stokes}} = \tilde\nu_{\text{abs,max}} - \tilde\nu_{\text{em,max}},
(12.1)

usually reported in cm−1\text{cm}^{-1} or eV so it can be compared across spectral regions. Also report the two peak wavelengths. Individual photons can overlap or even appear on the anti-Stokes side; the difference of band maxima is an operational measurement, not a fixed energy lost by every photon.

The mirror-image rule

For comparable potential shapes and the same emitting species, the lowest-energy absorption band and corrected emission band may look roughly mirror symmetric on an energy or wavenumber axis. Strong solvent effects, overlapping electronic bands, reabsorption, or different chemical forms can spoil this approximation. Estimate a 0–0 origin from mirror symmetry only if those conditions and the spectral overlap support it; otherwise report the observed peak separation without an invented origin.

Beer–Lambert

Light of intensity I0I_0 entering a sample of path length ℓ\ell and concentration cc emerges with

I=I0 10−εcℓ,A≡log⁡10I0I=εcℓ,I = I_0\,10^{-\varepsilon c \ell}, \qquad A \equiv \log_{10}\frac{I_0}{I} = \varepsilon c \ell ,
(12.2)

where AA is the absorbance and ε\varepsilon the molar absorptivity, in M−1cm−1\text{M}^{-1}\text{cm}^{-1}. Linear AA versus cc requires a stable absorbing species, known path length, and a detector operating above its dark and stray-light floor. Chemical changes, aggregation, scattering, and instrumental stray light can produce different deviations; the concentration series tests the range over which the relation holds.

Pre-lab

Apparatus

Panel (a), a shielded excitation LED illuminates a sample cuvette from the side; fluorescence travels at right angles through a matched long-pass filter to a detector spectrometer. Panel (b), a white lamp sends light through a sample cuvette to the spectrometer; a solvent-only cuvette supplies the reference spectrum.

Figure 12.2:Two geometries, one cuvette. (a) Excitation at 90 degrees to detection, with a long-pass filter, keeps the weak fluorescence from being swamped by scattered excitation light. (b) In-line transmission gives the absorption spectrum.

Procedure

Before you prepare the dilution series

Part A — Spectrometer setup

  1. Record a dark spectrum (source blocked) and a reference spectrum (solvent-only cuvette in the beam). Every absorbance is computed from these two, so retake them whenever the geometry or integration time changes.

  2. Verify wavelength calibration against a suitable known narrow line; an uncalibrated LED peak is not a wavelength standard.

  3. Confirm that both reference and sample signals stay below detector saturation and above the measured dark/stray-light floor over the wavelengths used for fitting.

[ ] Checkpoint 1. Show the instructor your dark and reference spectra, and a solvent-only “absorbance” spectrum, which should be flat and near zero. A sloping baseline here will masquerade as a spectral feature later.

Part B — Absorption and Beer–Lambert

  1. Prepare at least six known concentrations. Use the expected absorptivity and measured signal floor to aim for several peak absorbances from about 0.05 to 1.5; include a more concentrated sample only if useful for investigating departures. Record actual volumes and their uncertainties. Serial dilutions share stock and transfer uncertainties.

  2. Record the absorbance spectrum of each. Note the peak wavelength and peak absorbance.

  3. Plot peak absorbance against concentration and inspect residuals from the low-concentration line. Check whether departures follow the instrument’s dark or stray-light floor, scattering, a pH change, or a reproducible spectral-shape change. A moving peak alone does not prove aggregation.

Part C — Emission

  1. Choose a dilute sample, ideally Aexc≲0.1A_{\rm exc}\lesssim0.1 across the excitation path, and check absorbance across the emission band. At Aexc=0.1A_{\rm exc}=0.1, about 79% of incident light traverses a 1 cm path; stronger absorption changes where fluorescence originates. Reabsorption of emitted light can also alter its profile: together these are inner-filter effects.

  2. Illuminate from the side, at 90°90° to the collection axis, so that unabsorbed excitation light does not enter the spectrometer.

  3. Put the matched long-pass filter before the collection fiber and verify that it passes the emission peak rather than clipping the blue side.

  4. Record a dark and a solvent-only fluorescence blank at the same settings and subtract both appropriately. Check residual excitation leakage, Raman features, and filter or solvent background.

  5. If a second source also excites the same dye, repeat with its matched filter and blank. Compare normalized, background-corrected band shapes over their common passband. Kasha’s rule concerns relaxation to the lowest excited state; different apparent spectra can also arise from filters, detector response, or different chemical forms.

Part D — Concentration and quenching

  1. Record emission spectra across the dilution series at fixed excitation power, filter, and geometry. Plot background-corrected integrated emission against concentration. Expect an approximately linear low- concentration region if detector response and quantum yield are stable; a plateau or downturn at higher concentration can reflect inner-filter effects, quenching, aggregation, or detector saturation. Use dilution and geometry checks to distinguish them.

Part E (optional) — Chlorophyll

  1. If a prepared chlorophyll extract and approved solvent-handling setup are available, record its absorption and emission. Compare only an assigned absorption/emission pair, accounting for overlapping pigments, reabsorption, and solvent effects before interpreting the shift.

Analysis

Beer–Lambert

import numpy as np
from scipy.optimize import curve_fit

A  = np.array([...])           # peak absorbance
sA = np.array([...])
c  = np.array([...])           # molarity
sc = np.array([...])
ell_cm, sell_cm = 1.00, 0.01   # replace with measured path and uncertainty

sel = np.array([...], dtype=bool)  # select a range after checking floor and residuals
popt, pcov = curve_fit(lambda c, A0, m: A0 + m * c,
                       c[sel], A[sel], sigma=sA[sel], absolute_sigma=True)
A0, m = popt
eps = m / ell_cm
seps = np.hypot(np.sqrt(pcov[1, 1]) / ell_cm,
               abs(eps) * sell_cm / ell_cm)

Plot all points and fit a justified range whose transmission remains above the measured floor. Show the excluded points and residuals. A significant nonzero intercept calls for a blank or baseline check; do not silently force it to zero. The code propagates fit and path-length uncertainty but treats concentrations as exact. If their uncertainties matter, fit with a method that includes uncertainty in both axes or propagate the common stock and dilution errors by simulation. Report ε\varepsilon with its uncertainty and compare with a literature value for the same dye form, solvent, and pH.

Stokes shift and the mirror line

Convert both wavelength axes to wavenumber. Absorbance is dimensionless at each wavelength, so relabel its axis without multiplying the ordinates. For an emission spectrum calibrated as spectral density per nm, preserve area when changing axes: Fν~=Fλ ∣dλnm/dν~∣=Fλλnm2/107F_{\tilde\nu}=F_\lambda\,|d\lambda_{\rm nm}/d\tilde\nu| =F_\lambda\lambda_{\rm nm}^2/10^7 for ν~\tilde\nu in cm−1\text{cm}^{-1}. Raw counts per detector pixel require a wavelength-bin calibration before using that Jacobian. Correct the detector’s wavelength response and remove background before comparing band shapes or quoting a precise peak shift.

nu_abs = 1e7 / lam_abs_nm                # cm^-1; A_abs stays unchanged
nu_em  = 1e7 / lam_em_nm
F_nu   = F_lam_per_nm * lam_em_nm**2 / 1e7

Locate each band maximum using a documented local fit or smoothing window that is wider than the noise but narrower than the band; compare reasonable window choices. Report Δν~Stokes\Delta\tilde\nu_{\text{Stokes}} with uncertainty from calibration, noise, and peak-finding choices.

If an isolated lowest-energy absorption band and corrected emission band have sufficient overlap, reflect one about a trial wavenumber and test whether a plausible mirror line ν~00\tilde\nu_{00} aligns their resolved features. State the fit interval and sensitivity to background correction. If the bands do not support this test, report that limit. Half the Stokes shift equals the relaxation energy on either side only in the ideal equal-curvature displaced-harmonic model pictured above.

A vibrational quantum

If your spectra show a resolved, assigned vibronic progression, measure successive peak spacings. A nearly constant spacing may estimate the coupled mode’s vibrational quantum ℏω\hbar\omega; report it in cm−1\text{cm}^{-1} and meV. Compare it with kBTk_BT: the ground vibrational level dominates only when that mode’s spacing is large compared with kBTk_BT.

If no structure is resolved, state the instrument resolution and observed band width. Overlapping modes and broadening can hide even widely separated quanta, so an unresolved band alone gives no bound on their spacing.

Post-lab questions

Going further

Experiments
11. Alkali Spectra
Experiments
13. Counting Statistics and Half-Life