Einstein's Photoelectric Equation

In 1905 — his annus mirabilis, the year in which he also published the special theory of relativity and his paper on Brownian motion — Albert Einstein wrote a short paper entitled On a Heuristic Point of View Concerning the Production and Transformation of Light. In it he proposed a simple but revolutionary idea: light consists of discrete quanta of energy hf, and these quanta interact with matter one at a time.

Applied to the photoelectric effect, this hypothesis gave what is now called the Einstein photoelectric equation. In one line it resolves every puzzle Lenard had raised. It was for this result, not for relativity, that Einstein was awarded the Nobel Prize in Physics in 1921.

This lesson presents Einstein's equation, explains the physical meaning of each term, and shows how it accounts for all four observations of the previous lesson. The equation is central to the OCR A-Level Physics A specification (H556), Module 4.5.

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The Einstein Hypothesis

Einstein's starting assumption is simple. When light of frequency f is absorbed at a metal surface, the absorption happens one photon at a time. Each photon delivers its entire energy hf to a single electron, and the photon ceases to exist. There is no gradual "soaking up" of wave energy; there is no sharing of the photon's energy among many electrons. It is a single, discrete, all-or-nothing event.

This is the crucial conceptual leap. Classical wave theory pictures energy being absorbed smoothly and shared among many electrons. Einstein's picture is utterly different: one photon in, one excited electron out, total transfer hf.

Once this principle is accepted, the rest follows by simple energy conservation.

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The Work Function

Some of the energy delivered by the photon is needed just to liberate the electron from the metal. Conduction electrons in a metal are not free — they are held inside the metal by the combined electrostatic attraction of the positive ions and the potential well formed by the crystal lattice. To escape the metal surface, an electron must do work against this binding.

Definition. The work function φ of a metal is the minimum energy required to release an electron from the metal surface.

It is an intrinsic property of the metal (and of the cleanliness of the surface). Typical values are:

Metal	Work function φ (eV)	Work function (J)
Caesium	2.1	3.4 × 10⁻¹⁹
Potassium	2.3	3.7 × 10⁻¹⁹
Sodium	2.4	3.8 × 10⁻¹⁹
Zinc	4.3	6.9 × 10⁻¹⁹
Copper	4.7	7.5 × 10⁻¹⁹
Platinum	6.4	1.0 × 10⁻¹⁸

Notice how much these vary. Caesium and potassium have small work functions, which is why they are used in photocells for visible-light photocathodes. Copper's work function corresponds to a UV wavelength, which is why ordinary visible light cannot liberate electrons from a clean copper surface.

The work function φ is an average (or more precisely, a minimum) — electrons deep inside the metal are bound more strongly, but electrons just at the surface can escape with an energy equal to φ.

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Deriving the Einstein Equation

Consider a photon of frequency f striking a metal surface. It carries energy hf. When the photon is absorbed by an electron:

Energy in = Energy out
hf = (work to liberate electron) + (kinetic energy of freed electron)

An electron right at the surface needs only the minimum energy φ to escape, leaving the maximum possible kinetic energy:

hf = φ + KE_max

This is the Einstein photoelectric equation. In the form given in the OCR data sheet and most A-Level textbooks:

hf = φ + (1/2)mv²_max

where m is the electron mass and v_max is the maximum speed of the emitted photoelectrons.

Rearranging:

KE_max = hf - φ

This single line accounts for all four photoelectric observations.

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The Threshold Frequency

Setting KE_max = 0 in the Einstein equation gives the minimum frequency at which emission is just possible:

hf₀ = φ
f₀ = φ/h

Below this frequency, the photon does not carry enough energy even to overcome the work function — no emission occurs, and the photon's energy is simply converted to heat in the metal. At f = f₀, electrons are just barely liberated with negligible KE. Above f₀, the excess energy hf - φ appears as kinetic energy.

For caesium (φ = 2.1 eV = 3.4 × 10⁻¹⁹ J):

f₀ = φ/h = (3.4 × 10⁻¹⁹)/(6.63 × 10⁻³⁴) = 5.1 × 10¹⁴ Hz

The corresponding threshold wavelength:

λ₀ = c/f₀ = (3.00 × 10⁸)/(5.1 × 10¹⁴) = 590 nm

Yellow light. So yellow or any shorter-wavelength (higher-frequency) light can liberate electrons from caesium, but red light (650 nm) cannot.

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Explaining the Four Observations

Einstein's equation explains every feature of the photoelectric effect with trivial ease:

Threshold frequency. If f < f₀ = φ/h, then hf < φ and a single photon cannot supply enough energy to free an electron. No matter how many photons arrive — i.e. how intense the light — none of them individually has the energy needed. No emission. Above f₀, even a single photon is sufficient; emission begins immediately.

Instantaneous emission. A single photon liberates a single electron. There is no need to accumulate energy slowly from a wave. As soon as a photon arrives and is absorbed, the electron is freed.

Independence of KE_max from intensity. Each photon individually has energy hf, regardless of how many photons are present. More intensity just means more photons per second, which means more photoelectrons per second — it does not change the energy delivered to each individual electron. KE_max = hf - φ depends only on f.