Higher-order modes & the escape-factor's accidental success

Layer 1 showed that the Holstein equation has eigenmodes $\psi_j(\mathbf{r})$ each with its own trapping factor $g_j$. The lowest mode $\psi_0$ dominates at late times — so the "trapping factor" usually means $g_0$, and the escape-factor approximation (which gives just one number) often coincides with $g_0$.

That coincidence is the source of decades of subtle error. Whether $g_0$ is enough depends on how the vapour is being excited. If the excitation function $E(\mathbf{r})$ projects mostly onto $\psi_0$, then yes — the higher modes are irrelevant, $g_0$ is the answer, the escape factor is fine. But if the excitation looks like a narrow laser beam, or a wall-localized skin, or a δ-function at the cell centre, then the higher modes carry most of the photons and they decay much faster than $g_0\tau$.

This is why every modern fluorescent-lamp paper since the 1960s got away with the escape factor: electron-impact excitation in a cylindrical lamp has an approximately parabolic radial profile, and that parabola is essentially identical to the lowest Holstein mode $J_0(\zeta_0 r/R)$ in shape. The escape factor works in lamps not because the math is right but because the geometry happens to project the excitation onto $\psi_0$.

Explore

Pick an excitation profile. Watch the projection $\alpha_j$ of $E(\mathbf{r})$ onto each eigenmode. Then watch the emergent-intensity time profile $Y(t)$ — early-time decay is fast (high modes), late-time is slow ($g_0$). Toggle the number of retained modes; see when one mode is enough and when it lies to you.

The math

The initial excited-state distribution $n(\mathbf{r}, 0)$ is expanded into Holstein eigenmodes:

\[ n(\mathbf{r}, 0) = \sum_j \alpha_j \psi_j(\mathbf{r}), \qquad \alpha_j = \int_V n(\mathbf{r}, 0)\,\psi_j(\mathbf{r})\,d\mathbf{r} \]

Each mode decays with its own time constant $g_j\tau$:

\[ n(\mathbf{r}, t) = \sum_j \alpha_j \psi_j(\mathbf{r}) \exp\!\left(-\frac{t}{g_j \tau}\right) \]

The emergent intensity (photons per unit time leaving the cell) is then

\[ Y(t) = \frac{1}{\sum_j \hat\alpha_j g_j\tau} \sum_j \hat\alpha_j \exp\!\left(-\frac{t}{g_j\tau}\right), \qquad \hat\alpha_j = \frac{\alpha_j}{g_j}\int_V \psi_j(\mathbf{r})\,d\mathbf{r} \]

(M&O Eqs 4.12–4.14, 4.29.) Each mode contributes a pure exponential. The slowest ($g_0$) dominates at late $t$. The faster modes carry most of the early-time signal — and they're invisible if you only retain $j = 0$.

Why this layer matters

If you take exactly one thing from Layer 2: a single-exponential fit to a fluorescence-decay measurement is biased. The fit returns the slowest decay time, which (under trapping) is $g_0\tau$, not $\tau$. To recover $\tau$ from a vapour-cell measurement you must either (a) extrapolate to $k_0L \to 0$ (M&O Sec 15.1 method), (b) fit a multi-mode decay with at least 3 components, or (c) excite uniformly enough that only $\alpha_0$ is non-negligible.

Most published "natural lifetimes" of alkali resonance states in the 1960s–1980s did none of these. The literature is mildly polluted as a result. See M&O Sec 15.1 for the specific cleanups.

What this layer does NOT do

• The eigenfunctions $\psi_j$ for $j \ge 1$ in slab geometry use a crude $\zeta_j \approx \zeta_0 + j\pi$ approximation (M&O Eq 7.7 transcendental relation deferred). Adequate for visualisation; not for high-precision fits. Layer 4 will fix this when hyperfine splitting demands it.
• No Monte Carlo reference simulation — the 10-mode "converged" baseline is from the same fitting equations. A true MC cross-check requires precomputed data; see refs/cretin/ for analogues.
• No saturation (Layer 5), no PFR (Layer 3), no hfs (Layer 4).

FEYNMAN ✓ worked example · Victor ✓ five sliders · Kay ✓ reuses Lineshape + Geometry + Excitation objects from Layer 1 · Scott ⏳ MC reference cross-check deferred · Molisch ✓ Eqs 4.12–4.14 / 4.29 cited