Ionisation and the photoelectric effect

Result

Ionisation and the photoelectric effect

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Statement

When light of frequency \(f\) strikes a material with work function \(\phi\) (the minimum energy needed to remove an electron), the maximum kinetic energy of any ejected electron is \(KE_{\max}=hf-\phi\), and no electrons are ejected at all if \(f\) is below the threshold frequency \(f_0=\phi/h\), no matter how intense the light. For an isolated gas-phase atom, the same relation holds with the work function replaced by the atom's first ionisation energy \(I_1\): photoionisation occurs only for \(hf\ge I_1\).

Why it matters

This result is the experimental result that forced physics to accept that light itself is quantised, not just the energy levels of matter. Every other quantum idea in this unit — the Bohr model's discrete orbits, the Schrödinger equation's discrete quantum numbers — concerns the quantisation of an electron's energy. The photoelectric effect showed that the light doing the exciting is itself made of discrete packets, each carrying a fixed energy \(hf\); Einstein's 1905 paper on this effect (not his relativity papers) is what won him the 1921 Nobel Prize.

For chemistry specifically, the same equation, applied to gas-phase atoms rather than a bulk metal surface, is the operating principle of photoelectron spectroscopy (PES): shining monochromatic light of known, high energy on a sample and measuring the kinetic energy of the ejected electrons directly measures the ionisation energy of whichever orbital each electron came from — the experimental technique that produced the accurate ionisation-energy data this unit's periodic-trends result is checked against.

Hypotheses

Light is treated as a stream of photons, each of energy \(hf\), not a continuous classical wave.The classical (pre-1905) wave picture of light predicts something different: a wave's energy is spread continuously over its wavefront, so a dim (low-intensity) beam of any frequency should eventually accumulate enough energy at a given point to eject an electron, given enough exposure time, and \(KE_{\max}\) should increase with intensity (a "bigger wave" delivering more energy), not frequency. Neither prediction matches experiment: below the threshold frequency, no electrons are ejected even after hours of exposure to intense light, and \(KE_{\max}\) is observed to depend only on frequency, never on intensity — exactly opposite to the classical wave prediction, and exactly as the photon (particle) picture in the Statement predicts. One photon interacts with (at most) one electron.At extremely high light intensities (achievable only with pulsed lasers, well beyond ordinary laboratory sources), a single electron can absorb two or more photons nearly simultaneously (multiphoton photoemission), and the simple single-photon formula \(KE_{\max}=hf-\phi\) no longer applies on its own. The result as stated is the ordinary, single-photon regime that accounts for the vast majority of photoelectric and photoionisation experiments, including all standard photoelectron spectroscopy.

Proof

E_{\text{photon}} = hf

Einstein's 1905 postulate (building on Planck's 1900 quantisation of blackbody radiation, but extending it — radically, for its time — to light itself, not just to the oscillators emitting it): light of frequency \(f\) is absorbed and emitted only in discrete quanta of energy \(hf\), later named photons. This is the one new physical assumption in the whole derivation. A

\text{Energy conservation for one electron absorbing one photon: } hf = \phi + KE

The absorbed photon's energy must go somewhere: part of it, \(\phi\), pays the minimum binding energy holding the electron in the material (or atom); whatever energy remains becomes the ejected electron's kinetic energy. A

KE_{\max} = hf - \phi

Solve Step 2 for \(KE\). The maximum kinetic energy corresponds to an electron that was bound by exactly \(\phi\) (the minimum possible binding, i.e. the least tightly held electron available); any more tightly bound electron loses more energy paying its larger effective binding cost and emerges with less kinetic energy, so \(KE_{\max}\) is a strict upper bound on the ejected-electron energies observed at that frequency. A

KE_{\max}\ge0 \ \Longrightarrow\ hf\ge\phi \ \Longrightarrow\ f\ge\frac{\phi}{h}\equiv f_0

Kinetic energy cannot be negative, so Step 3 forces a minimum photon frequency \(f_0=\phi/h\) below which \(KE_{\max}\) would have to be negative — physically impossible, meaning no electron is ejected at all. This is the threshold-frequency phenomenon Hypotheses flagged as inexplicable classically: it falls out immediately from the photon picture, with no extra assumption needed beyond Step 1. A

Result

KE_{\max} = hf - \phi, \qquad f_0=\frac{\phi}{h} \qquad (\text{no emission below } f_0, \text{ regardless of intensity})

Reading. One photon in, at most one electron out, with the photon's energy split between overcoming binding and leaving as kinetic energy; a frequency threshold is an automatic, unavoidable consequence, not a separate empirical add-on.

Scope. Applies to bulk-material photoemission (with \(\phi\) the material's work function) and to gas-phase photoionisation of atoms or molecules (with \(\phi\) replaced by the relevant ionisation energy) alike, in the ordinary single-photon-absorption regime.

Corollaries & converses

A plot of \(KE_{\max}\) versus \(f\) is a straight line of slope exactly \(h\) (Planck's constant) and \(x\)-intercept \(f_0=\phi/h\) — Millikan's painstaking 1916 experimental verification of this exact linear relationship, including measuring \(h\) from the slope and finding it matched Planck's blackbody value, was the decisive confirmation of Einstein's photon hypothesis.
In photoelectron spectroscopy, measuring the kinetic energy of ejected electrons at a known, fixed photon energy directly gives the ionisation energy of the orbital they came from, via \(I=hf-KE\) — the rearranged Result, run in the direction of measuring \(\phi\) (here, \(I\)) rather than predicting \(KE\).
Converse: observing electron ejection at some frequency \(f\) guarantees \(f\ge f_0\), but observing no ejection does not by itself distinguish "below threshold" from "photon flux too low to detect a small photocurrent" — the threshold claim specifically concerns whether emission occurs at all, in principle, not whether it is experimentally detectable at a given intensity.

Fails without

Drop the photon (single-quantum) picture, keep classical waves: classical electromagnetic theory predicts that even a very low-frequency wave, given enough time, delivers enough accumulated energy at a point to eject an electron — predicting a time delay before emission that grows as intensity falls, and no sharp threshold frequency at all. Careful photoelectric experiments (notably Lenard's, pre-1905, and later Millikan's precision measurements) found emission is essentially instantaneous (within nanoseconds) at any frequency above threshold, at any intensity, and strictly absent below threshold regardless of exposure time — flatly contradicting the classical wave prediction.
Drop energy conservation (Step 2): without it there is no reason \(KE_{\max}\) should depend on \(f\) at all; the entire derivation collapses to nothing more than "light ejects electrons," with none of the quantitative, checkable content (the linear \(KE_{\max}\)-vs-\(f\) relationship, the specific slope \(h\)) that made the result experimentally decisive.

Common errors

Assuming brighter (more intense) light ejects electrons with more kinetic energy. Intensity increases the number of photons per second (hence the photocurrent, the rate of electron ejection), not the energy of each individual photon; only frequency sets \(KE_{\max}\).
Forgetting the threshold condition and computing a negative \(KE_{\max}=hf-\phi\) as if it were physically meaningful. A negative result means no electrons are ejected at that frequency at all — the formula's regime of validity is \(f\ge f_0\) only.
Confusing the work function \(\phi\) (a bulk material property, typically a few eV, roughly independent of which specific electron is removed) with the first ionisation energy \(I_1\) of an isolated gas-phase atom of the same element; the two are related but numerically different concepts — the work function is generally somewhat lower than the corresponding gas-phase ionisation energy, because a delocalised electron near a solid's surface is, on average, less tightly bound than one in an isolated atom.

Discussion

The effect itself was first observed by Heinrich Hertz in 1887 (as a side effect while confirming Maxwell's electromagnetic waves — sparks jumped more readily under ultraviolet illumination) and investigated systematically by Philipp Lenard around 1902, who established the puzzling experimental facts (frequency-dependent threshold, intensity-independent \(KE_{\max}\)) that had no classical explanation. Einstein's 1905 paper proposed the photon explanation as one of several radical ideas in his "miracle year," alongside special relativity and Brownian motion — and it was this paper, not relativity, that the Nobel committee cited in 1921, in part because Robert Millikan's decade-long, extremely careful experimental campaign (completed 1916) had by then verified the linear \(KE_{\max}\)-vs-\(f\) relationship and the value of \(h\) to high precision, giving the Nobel committee unambiguous experimental grounds that relativity, still viewed cautiously by some physicists at the time, did not yet have.

Millikan himself later wrote that he had set out to disprove Einstein's "reckless" photon hypothesis and instead spent years confirming it in exacting quantitative detail — a well-known episode in the history of physics illustrating how a theory's own strongest early critic can become its most rigorous experimental verifier.

The photon picture here is a special case of a more general principle: any process in which energy is absorbed or emitted in discrete quanta of size \(hf\) (or, more generally, \(\hbar\omega\)) reproduces this same threshold-and-linear-slope signature. The same underlying logic explains, for example, the threshold behaviour of Compton scattering and, further afield, the discrete-energy-loss spectra seen in inelastic electron scattering off atoms (the Franck–Hertz experiment) — different experiments, the same "quanta come in fixed lumps, not a continuous trickle" signature.

Common misconception: that the photoelectric effect "proves light is a particle" in some final, absolute sense, superseding the wave picture entirely. It does no such thing: interference and diffraction (unambiguously wave phenomena) remain fully valid and are observed under different experimental conditions with the very same light. The photoelectric effect shows that light's energy exchange with matter is quantised; wave–particle duality, not a simple particle victory, is the correct and complete resolution, developed further once de Broglie extended the same idea to matter.

Worked examples

\text{Sodium, } \phi=2.28\,\text{eV}: \quad f_0=\frac{\phi}{h}, \qquad \lambda_0=\frac{hc}{\phi}\approx543.8\,\text{nm}

Threshold wavelength (converting via \(c=f\lambda\)) for sodium metal; \(543.8\,\text{nm}\) sits in the blue-green part of the visible spectrum, meaning sodium photoemits under blue or violet light and shorter wavelengths, but not under red or infrared light, however intense. A

200\,\text{nm UV light on sodium}: \quad E_{\text{photon}}=\frac{hc}{\lambda}\approx6.20\,\text{eV}, \qquad KE_{\max}=6.20-2.28\approx3.92\,\text{eV}

\(200\,\text{nm}\) is well above sodium's threshold, so emission occurs; the ejected electrons' maximum kinetic energy follows directly from the Result. This kind of calculation is standard in photoelectron spectroscopy: the photon energy is fixed by the light source, the ejected electron's \(KE\) is measured, and \(\phi\) (or, for a gas-phase atom, the ionisation energy) is read off by subtraction. A

\lambda_0(\text{Na})\approx543.8\,\text{nm}; \quad KE_{\max}(200\,\text{nm on Na})\approx3.92\,\text{eV}

Reading. A single work function, together with the Result, predicts both the threshold colour of light at which sodium begins to photoemit and exactly how energetic the ejected electrons are for any higher-energy light used.

Scope. The identical two-step calculation applies to any material with known \(\phi\), and to any gas-phase atom with known ionisation energy, using whichever photon wavelength is of interest.

Problems

Platinum has a work function of \(6.35\,\text{eV}\), among the highest of any metal. Does \(300\,\text{nm}\) ultraviolet light eject photoelectrons from platinum? Justify with a calculation.
Solution
Photon energy at \(300\,\text{nm}\): \(E=hc/\lambda\approx1239.84/300\approx4.13\,\text{eV}\). Since \(4.13\,\text{eV}<6.35\,\text{eV}=\phi\), the photon energy is below platinum's work function — no photoelectrons are ejected, regardless of how intense the \(300\,\text{nm}\) beam is made. Platinum's unusually high work function is why it is sometimes used as a reference "non-responsive" electrode in photoelectric demonstrations using visible or near-UV light.
An ejected photoelectron is found to have \(KE_{\max}=1.50\,\text{eV}\) when illuminated at \(f=1.20\times10^{15}\,\text{Hz}\). Find the work function of the material.
Solution
Photon energy: \(E=hf\). Using \(h=4.136\times10^{-15}\,\text{eV}\cdot\text{s}\) (Planck's constant in eV-friendly units), \(E=4.136\times10^{-15}\times1.20\times10^{15}\approx4.96\,\text{eV}\). From the Result, \(\phi=hf-KE_{\max}=4.96-1.50=3.46\,\text{eV}\).
Explain why measuring the stopping voltage (the retarding potential just sufficient to stop the most energetic ejected electrons) is the standard experimental method for measuring \(KE_{\max}\), rather than measuring electron speeds directly.
Solution
An electron of charge \(e\) moving against a retarding potential difference \(V_{\text{stop}}\) loses kinetic energy \(eV_{\text{stop}}\) by the time it (just) comes to rest; setting this equal to \(KE_{\max}\) gives \(eV_{\text{stop}}=KE_{\max}\), i.e. \(V_{\text{stop}}=KE_{\max}/e\) (numerically, \(V_{\text{stop}}\) in volts equals \(KE_{\max}\) in eV). Measuring the voltage at which the photocurrent just drops to zero is a precise, purely electrical measurement, avoiding the practical difficulty of measuring individual electron speeds directly — exactly the method Millikan used to establish the linear \(KE_{\max}\)-vs-\(f\) relationship to high precision.
A gas-phase argon atom (first ionisation energy \(I_1=15.76\,\text{eV}\)) is illuminated with \(58.4\,\text{nm}\) light (a common helium-discharge-lamp line used in photoelectron spectroscopy). Determine whether photoionisation occurs, and if so, find the kinetic energy of the ejected electron.
Solution
Photon energy: \(E=hc/\lambda\approx1239.84/58.4\approx21.23\,\text{eV}\). Since \(21.23\,\text{eV}>15.76\,\text{eV}=I_1\), photoionisation occurs (the gas-phase analogue of the Statement, with \(I_1\) playing the role of \(\phi\)). Kinetic energy of the ejected electron: \(KE=E-I_1=21.23-15.76\approx5.47\,\text{eV}\) — this is exactly the kind of measurement used in ultraviolet photoelectron spectroscopy (UPS) to determine gas-phase ionisation energies experimentally.