The Photoelectric Effect Explained: E = hf Minus the Work Function

The photoelectric effect is the emission of electrons from a material, usually a metal, when light shines on it. Einstein's 1905 explanation says light arrives in quanta of energy hf, and an electron escapes only if one quantum supplies at least the work function φ, so the fastest electrons carry kinetic energy K_max = hf minus φ. Below the threshold frequency f₀ = φ/h no electrons leave, however bright the light.

That one line of algebra ended a long argument about what light is, earned Einstein the 1921 Nobel Prize in Physics, and earned Robert Millikan part of the 1923 prize for testing it. It also sits underneath photomultiplier tubes, the spectrometers that map electrons in solids, and, in a related form, the image sensor in your phone and the solar panel on a roof.

This post goes one level below our tour of famous equations that changed the world and pairs with E = mc² explained, which covers Einstein's other famous 1905 result. For where ultraviolet, visible and X-ray photons sit on the frequency scale, see the electromagnetic spectrum explained. For the hardware side, read how camera image sensors are made and how solar panels are made.

A note on numbers. The exact values of h, e and c come from the BIPM list of SI defining constants and NIST's CODATA 2022 tables. Work functions come from the HyperPhysics table at Georgia State University, which draws partly on the Handbook of Chemistry and Physics. Mercury spectral lines come from NIST's Handbook of Basic Atomic Spectroscopic Data. The history comes from the Nobel Foundation's own pages, the American Physical Society's archive of Millikan's 1916 paper and its centennial Focus article, and scans of the original Annalen der Physik volumes on the Internet Archive. Values we picked ourselves, such as a lamp's power or a detector's quantum efficiency, are labelled as illustrative. Every number below was recomputed with the short Python script in the walkthrough.

Key takeaways

• The equation: K_max = hf minus φ, where h is the Planck constant, f the light's frequency and φ the work function, the minimum energy needed to pull an electron out of that surface. In practical units, photon energy in electronvolts is 1239.84 divided by the wavelength in nanometres.
• Threshold, not brightness: emission starts at f₀ = φ/h, or below the threshold wavelength λ₀ = hc/φ. Brighter light above threshold releases more electrons per second, not faster ones. Brighter light below threshold releases none.
• History in four dates: Heinrich Hertz reported in 1887 that ultraviolet light helped a spark jump a gap. Philipp Lenard reported in 1902 that the electrons' speed did not depend on intensity. Einstein proposed light quanta in Annalen der Physik volume 17 in 1905. Millikan's paper in Physical Review volume 7, published 1 March 1916, confirmed the straight line and measured h.
• Nobel wording: Einstein's 1921 prize cited "especially his discovery of the law of the photoelectric effect", not relativity, and was presented in 1922. Millikan's 1923 prize was "for his work on the elementary charge of electricity and on the photoelectric effect."
• 2019 SI: h and e are now exact by definition, so h/e (the slope Millikan measured) and hc/e (the 1239.84 shortcut) are exact too. Work functions are still measured, and they vary with surface condition.
• Worked results: zinc (φ = 4.3 eV) needs light shorter than 288.3 nm. Mercury's 253.65 nm line ejects electrons from zinc with at most 0.588 eV, stopped by 0.588 V, moving at up to 455 km/s. Red 650 nm light does nothing to sodium, while 405 nm violet light frees electrons with up to 0.78 eV.
• Related but different: the external photoelectric effect sends electrons out of the material into vacuum. Photodiodes, CCD and CMOS pixels and solar cells use the internal version, where absorbed photons free charge carriers inside a semiconductor.

What is the photoelectric effect?

Shine light on a clean metal surface in a vacuum, put a second electrode nearby, and a small current flows between them. The light is knocking electrons out of the metal. Those emitted electrons are called photoelectrons, and the surface that emits them is a photocathode.

The Nobel Foundation's summary of Einstein's prize describes the effect the way it was first noticed: when metal electrodes are exposed to light, sparks between them jump more readily, but only if the light is above a certain frequency. The surprise in that last clause is the whole story. Classical physics treated light as a continuous wave whose energy depends on its intensity, so a bright enough beam of any colour should eventually shake electrons loose. Experiments said otherwise.

Three observations stood out once people measured the effect carefully:

1. There is a threshold frequency. For each surface, light below a certain frequency produces no photoelectrons at all, no matter how intense.
2. Electron speed depends on frequency, not intensity. Making the light brighter increases the number of electrons emitted per second but not their maximum kinetic energy. Raising the frequency raises the maximum kinetic energy.
3. There is no measurable delay. Emission starts as soon as light above threshold arrives, even when the light is dim. A wave model that spreads energy evenly over the surface predicted that an electron would need time to soak up enough energy.

The 1921 Nobel presentation speech singles out the second point: it calls it the most extraordinary aspect of the effect that the emission velocity is independent of the light's intensity, which sets only the number of electrons, while velocity rises with frequency.

What did Hertz see in 1887?

Heinrich Hertz found the effect by accident while doing something else. In "Ueber einen Einfluss des ultravioletten Lichtes auf die electrische Entladung" (On an effect of ultraviolet light on the electric discharge), published in Annalen der Physik und Chemie volume 31, starting on page 983, in 1887, he describes an experiment on resonance between fast electrical oscillations. One induction coil discharge drove two sparks at once: a bright primary spark A and a fainter secondary spark B, whose maximum length he was measuring.

When he put a darkening case around spark B to see it better, the maximum spark length got noticeably shorter. Removing the case piece by piece, he found that only the wall between spark A and spark B mattered, and it mattered even when moved well away from B. In the rest of the paper he traces the effect to ultraviolet light from spark A falling on the gap of spark B. In modern terms, ultraviolet photons were freeing electrons from the electrodes and making it easier for the discharge to start.

Hertz did not explain the mechanism. The electron itself was not identified until a decade later. The Nobel Foundation's 1921 presentation speech credits Hertz with the discovery in 1887 and says Wilhelm Hallwachs then studied it more thoroughly, showing that under certain conditions a negatively charged metal plate lit with ultraviolet light loses its negative charge and ends up positively charged. The same speech says Lenard showed in 1899 that the cause was the emission of electrons.

What did Lenard show, and why did it break wave theory?

Philipp Lenard, who won the 1905 Nobel Prize in Physics for his cathode ray experiments, measured the energies of the emitted particles. His 1902 paper in Annalen der Physik, fourth series, volume 8, page 149, is one of the references Millikan cites in his 1916 paper. The Nobel Foundation's 1921 presentation speech summarises the finding: the speed of the emitted electrons did not depend on the intensity of the light, and Lenard stressed that this did not fit the ideas of the time.

Here is why it is so hard for a classical wave. In a wave, the energy delivered per second per square metre is the intensity. Double the intensity and every electron near the surface is shaken twice as hard, so the most energetic escapees should be faster. Lenard's measurements showed more electrons at higher intensity but the same top speed. The table summarises what the wave picture predicted against what experiments found.

What did Einstein propose in 1905, and what is the equation?

In 1905 Einstein published "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" (On a heuristic point of view concerning the production and transformation of light) in Annalen der Physik, fourth series, volume 17, starting on page 132. The volume's table of contents on the Internet Archive lists the paper, and Millikan's 1916 paper cites it as "Ann. d. Phys. (4), 17, 132, 1905". In the same volume Einstein also published his paper on the motion of particles suspended in liquids, the Brownian motion work.

The core idea, in the Nobel Foundation's summary, is that light consists of quanta, packets with fixed energies tied to their frequency, and one such quantum must have a certain minimum frequency before it can free an electron. Max Planck had earlier introduced energy quanta to fix the theory of blackbody radiation, but he treated them as a property of how matter emits and absorbs. Einstein took the bolder step of treating light itself as made of quanta.

Applied to a metal, the argument goes like this. One quantum of energy hf is absorbed by one electron. Part of that energy is spent getting the electron out through the surface. Whatever is left becomes the electron's kinetic energy. The 1921 presentation speech describes exactly this bookkeeping. Electrons near the surface that lose nothing on the way out leave with the most energy:

K_max = hf minus φ

Three rearrangements of that one equation do almost all the work in practice.

Threshold. Set K_max to zero: f₀ = φ/h, and in wavelength terms λ₀ = hc/φ. Light with a longer wavelength than λ₀ cannot eject electrons.

Stopping potential. If you make the collector negative relative to the photocathode, you push back on the photoelectrons. The voltage that just stops the fastest ones satisfies eV₀ = K_max, so V₀ = (h/e)f minus φ/e. A plot of V₀ against f is a straight line with slope h/e, the same for every metal, and an intercept that depends on the metal. That straight line is the prediction Millikan set out to test.

The electronvolt shortcut. One electronvolt is the energy an electron gains crossing one volt, exactly 1.602 176 634 × 10⁻¹⁹ J. Photon energy in eV is then E = hc/(eλ), and hc/e works out to 1239.841 984... nm eV. So a photon's energy in eV is about 1239.84 divided by its wavelength in nanometres, and a threshold wavelength in nanometres is about 1239.84 divided by the work function in eV.

What is a work function, physically?

Electrons in a metal fill energy levels up to the Fermi level. To leave, an electron has to climb from there to the vacuum level just outside the surface. The work function is that climb. It is not a fixed atomic property like an ionisation energy. It depends on which crystal face is exposed, on oxide layers, on adsorbed gases and on contamination. HyperPhysics notes that polycrystalline silver measures 4.26 eV while the (111) face of a single crystal measures 4.74 eV, with the (100) and (110) faces at 4.64 and 4.52 eV. That spread is why careful experiments, from Millikan's to modern spectrometers, go to great lengths to prepare clean surfaces in vacuum.

A short timeline of the photoelectric effect

Worked example 1: threshold frequency and wavelength for zinc

Zinc is a good first metal because its threshold sits in the ultraviolet, which is exactly why Hertz's sparks needed ultraviolet light rather than lamplight. HyperPhysics lists zinc's work function as 4.3 eV.

Step 1: convert the work function to joules. φ = 4.3 eV × 1.602 176 634 × 10⁻¹⁹ J/eV = 6.889 × 10⁻¹⁹ J.

Step 2: threshold frequency. f₀ = φ/h = 6.889 × 10⁻¹⁹ J ÷ 6.626 070 15 × 10⁻³⁴ J s = 1.0397 × 10¹⁵ Hz.

Step 3: threshold wavelength. λ₀ = c/f₀ = 299 792 458 m/s ÷ 1.0397 × 10¹⁵ Hz = 2.883 × 10⁻⁷ m = 288.3 nm.

Check with the shortcut. 1239.84 ÷ 4.3 = 288.3 nm. The two routes agree, as they must, because the shortcut is just hc/e.

What it means. Visible light runs from roughly 380 to 750 nm, so every visible colour has a wavelength longer than 288.3 nm. No visible lamp of any brightness will eject electrons from clean zinc. You need ultraviolet light shorter than about 288 nm.

The same arithmetic for a low work function metal shows the other extreme. Cesium, at 2.1 eV in the HyperPhysics table, has λ₀ = 1239.84 ÷ 2.1 = 590.4 nm, in the yellow. Blue and green light free electrons from cesium; red light does not. That is why alkali metals and their compounds show up in photocathodes meant to respond to visible light.

Worked example 2: maximum electron energy and stopping potential for UV light on zinc

Now shine ultraviolet light from a mercury lamp on the same zinc surface. NIST's table of strong mercury lines lists a neutral mercury line at 2536.517 Å, which is 253.65 nm. (NIST gives these lines as wavelengths in air; the difference from vacuum is about 0.03 percent and changes nothing below.)

Step 1: frequency of the light. f = c/λ = 299 792 458 ÷ 253.6517 × 10⁻⁹ = 1.1819 × 10¹⁵ Hz. That is above zinc's threshold of 1.0397 × 10¹⁵ Hz, so emission happens.

Step 2: photon energy. E = hf = 6.626 070 15 × 10⁻³⁴ × 1.1819 × 10¹⁵ = 7.831 × 10⁻¹⁹ J = 4.888 eV. With the shortcut: 1239.84 ÷ 253.65 = 4.888 eV.

Step 3: maximum kinetic energy. K_max = hf minus φ = 4.888 eV minus 4.3 eV = 0.588 eV, or 9.42 × 10⁻²⁰ J.

Step 4: stopping potential. V₀ = K_max/e = 0.588 V. Set the collector 0.588 V negative relative to the zinc and the photocurrent falls to zero. Because K_max in electronvolts and V₀ in volts are numerically identical, many lab reports skip this step.

Step 5: maximum speed. K_max = ½mv², so v = √(2K_max/m_e) = √(2 × 9.42 × 10⁻²⁰ ÷ 9.109 × 10⁻³¹) = 4.55 × 10⁵ m/s, about 455 km/s. That is about 0.15 percent of the speed of light, so the non-relativistic formula is fine.

Sensitivity to φ. Zinc's work function is given to two significant figures, and real surfaces vary. If an oxidised sample had φ = 4.5 eV instead, K_max would drop to 0.388 eV and V₀ to 0.388 V. The uncertainty in the answer is dominated by the surface, not by the constants.

Worked example 3: why brightness cannot replace frequency

Take sodium (φ = 2.28 eV in the HyperPhysics table) and zinc (4.3 eV), and four common light sources: a red 650 nm laser pointer, a green 532 nm laser, a violet 405 nm laser diode and mercury's 365.02 nm ultraviolet line from the NIST table. The laser wavelengths are typical illustrative values.

Now make the red laser a thousand times brighter. Each photon still carries 1.907 eV, still 0.373 eV short of sodium's work function, and in the simple one photon picture nothing comes out. Make the violet laser a thousand times brighter and you get roughly a thousand times as many photoelectrons per second, each with the same 0.781 eV maximum. That is Lenard's observation in numbers, and it is the cleanest way to see why intensity and frequency play completely different roles.

(At extreme intensities, such as focused pulsed lasers, an electron can absorb two or more photons at once and escape below the single photon threshold. That multiphoton photoemission is real but needs intensities far beyond any lamp or pointer. The simple equation describes one photon per electron.)

Reverse problem: identify the metal. Suppose 300 nm light gives a measured stopping potential of 1.20 V. The photon energy is 1239.84 ÷ 300 = 4.133 eV, so φ = 4.133 minus 1.20 = 2.93 eV. In the HyperPhysics table, calcium is listed at 2.9 eV, the closest match. In practice you would never identify a metal this way, because surface conditions shift φ by tenths of an electronvolt, but it is a good exam question.

Worked example 4: a Millikan style plot that recovers h

Millikan's method was to measure the stopping potential at several frequencies on one surface and fit a line. The slope gives h/e and the intercept gives the work function. Here is an idealised version using sodium (φ = 2.28 eV) and six strong mercury lines from the NIST table. These are calculated, noise free values, not Millikan's measurements.

The green 546.07 nm line is the interesting one. Its photons carry 2.270 eV, just 0.01 eV short of sodium's 2.28 eV, so it produces nothing even though it is only 2.3 nm longer than sodium's 543.8 nm threshold.

Fit the five points that emit. A least squares line through (f, V₀) gives:

• slope = 4.135 668 × 10⁻¹⁵ V s, which is h/e
• h = slope × e = 4.135 668 × 10⁻¹⁵ × 1.602 176 634 × 10⁻¹⁹ = 6.626 07 × 10⁻³⁴ J s
• intercept = minus 2.280 V, so φ = 2.280 eV
• threshold frequency = intercept ÷ slope = 5.513 × 10¹⁴ Hz

With real data the points scatter, the intercept is pulled around by surface contamination and contact potentials between electrodes, and the slope is what survives. That is why Millikan could get h to within a percent while the work functions he found were less certain.

There is a historical echo in the data. HyperPhysics' page on early photoelectric data for sodium, which it links to Millikan, gives a threshold of 683 nm, or 1.82 eV, well below the 2.28 eV handbook value for clean sodium. Surface condition and the contact potential between electrodes shift the apparent threshold. The slope, h/e, is the same for every surface.

Worked example 5: from photons per second to photocurrent

The equation tells you the energy of each electron. How many come out depends on how many photons arrive and on the quantum efficiency, the fraction of photons that produce a photoelectron.

Suppose an illustrative 1.00 mW of 253.65 nm light falls on a photocathode.

Step 1: photon energy in joules. 7.831 × 10⁻¹⁹ J, from worked example 2.

Step 2: photons per second. N = P/E = 1.00 × 10⁻³ W ÷ 7.831 × 10⁻¹⁹ J = 1.277 × 10¹⁵ photons per second.

Step 3: photocurrent for a few quantum efficiencies. I = N × QE × e.

Double the light power and the current doubles; the stopping potential does not move. This is also why photon counting detectors are possible. One photon can release one electron, and a detector that multiplies that single electron can register single photons, which is what a photomultiplier tube does.

How did Millikan test the equation and measure Planck's constant?

Robert Millikan, then at the Ryerson Physical Laboratory of the University of Chicago, published "A Direct Photoelectric Determination of Planck's h" in Physical Review volume 7, page 355, on 1 March 1916. The American Physical Society's centennial Focus article (1999) summarises it: Millikan was the first to show with high accuracy that the maximum kinetic energy of photoelectrons obeys Einstein's 1905 equation, and he found h = 6.57 × 10⁻²⁷ erg seconds, which he stated was good to about 0.5 percent.

Two pieces of craft made that possible, according to the APS account.

• A clean surface in vacuum. Reactive metal surfaces tarnish quickly and contamination shifts the work function. Millikan built what he called a "machine shop in vacuo", a rotating knife driven electromagnetically from outside the evacuated vessel, that shaved a fresh surface on the metal just before he illuminated it with monochromatic light.
• A trusted value of e. Converting a stopping potential to an energy needs the electron's charge. Millikan used his own oil drop value. NobelPrize.org's facts page dates his precise determination of e to 1910, while the APS Focus article refers to the value he established in 1913; both describe the same oil drop programme at different stages.

Converted to SI units, 6.57 × 10⁻²⁷ erg s is 6.57 × 10⁻³⁴ J s. Today's exact value is 6.626 070 15 × 10⁻³⁴ J s, so Millikan's result was about 0.85 percent low, somewhat outside the 0.5 percent he quoted. That does not diminish the result: no earlier attempt came close, and the straight line with a universal slope was exactly what Einstein predicted.

The irony is well documented. The APS article notes that in a January 1916 paper in the same volume, Millikan wrote that Einstein's equation represented the data very accurately but could not yet be regarded as resting on a satisfactory theoretical foundation. In the h paper itself he called the light quantum "the bold, not to say the reckless, hypothesis". He confirmed the equation while doubting the physics behind it.

What were the 1921 and 1923 Nobel Prizes for?

The wording matters, because a common myth says Einstein won for relativity.

• Einstein, Physics 1921. NobelPrize.org gives the motivation as for his services to theoretical physics, and "especially for his discovery of the law of the photoelectric effect". The Nobel Committee decided in 1921 that no nomination met the criteria, reserved the prize, and awarded it a year later; it was announced on 9 November 1922. Einstein could not attend the ceremony. His Nobel lecture, given in Gothenburg on 11 July 1923, was on relativity, even though the prize citation was not.
• Millikan, Physics 1923. The motivation reads "for his work on the elementary charge of electricity and on the photoelectric effect". The presentation speech on 10 December 1923 put it bluntly: had Millikan's photoelectric results come out differently, Einstein's law would have been without value and Bohr's theory without support. Millikan's own Nobel lecture, delivered on 23 May 1924, was titled "The Electron and the Light-Quant from the Experimental Point of View".

The 1921 presentation speech also makes a practical claim worth noting: it says Einstein's law had become the basis of quantitative photochemistry, the way Faraday's law underpins electrochemistry.

What do the 2019 SI definitions fix exactly, and how does that change the arithmetic?

On 16 November 2018, at Versailles, the General Conference on Weights and Measures voted to redefine the International System of Units on fixed constants of nature. NIST reports the change took effect on 20 May 2019, redefining the kilogram, kelvin, ampere and mole. The BIPM lists seven defining constants and states that their numerical values have no uncertainty. Three of them appear in the photoelectric equation:

NIST notes that the value of h was only finalised in 2017 and that h now helps define the kilogram, the kelvin and the candela, while e helps define the ampere.

What changes in the arithmetic:

1. Derived ratios are exact. h/e = 4.135 667 696... × 10⁻¹⁵ V s, the slope of the stopping potential line, is now exact. So is hc/e = 1239.841 984... nm eV. NIST's CODATA page lists the Planck constant in eV/Hz as exact.
2. The electronvolt is an exact number of joules. 1 eV = 1.602 176 634 × 10⁻¹⁹ J exactly. Converting a work function from eV to J adds no uncertainty.
3. Millikan's experiment flips meaning. In 1916, measuring the slope h/e was a way to determine h. Today h/e is fixed, so the same experiment in a teaching lab checks the apparatus, the light source and the analysis rather than measuring a constant of nature.
4. Uncertainty moves to the material. Every bit of uncertainty in a photoelectric calculation now comes from the work function, the wavelength measurement and the surface. Constants that are still measured, such as the electron mass (9.109 383 7139 × 10⁻³¹ kg in CODATA 2022, with a relative standard uncertainty of 3.1 × 10⁻¹⁰), only enter when you convert energy into speed.
5. Older numbers drift slightly. Textbooks printed before 2019 often use h = 6.626 × 10⁻³⁴ J s and e = 1.602 × 10⁻¹⁹ C, or the shortcut 1240 nm eV. Those roundings change answers in the fourth significant figure, well below the spread in real work functions.

How is the photoelectric effect different from the photovoltaic effect and photodiodes?

They share the same first step, a photon giving its energy to an electron, and differ in where the electron ends up.

In the external photoelectric effect, the one Einstein explained, the electron leaves the material entirely and travels through vacuum or gas to a collector. The energy barrier is the work function, a few electronvolts for metals.

In the internal photoelectric effect, the electron never leaves the solid. In a semiconductor, a photon with enough energy lifts an electron from the valence band into the conduction band, leaving a hole behind. The barrier is the band gap rather than the work function. The NobelPrize.org press release for the 2009 prize, which honoured Willard Boyle and George Smith for inventing the CCD sensor in 1969, says the CCD makes use of the photoelectric effect as Einstein theorised it, turning light into electric signals. That is the internal effect: each pixel collects the charge freed inside the silicon.

The photovoltaic effect adds a built in electric field. The US Department of Energy explains that a PV cell absorbs sunlight and transfers the energy to electrons in the semiconductor, and those charges then move in response to an internal electric field in the cell, causing current to flow out through the metal contacts. No external voltage is needed; the cell is a power source. DOE notes a single cell typically produces about 1 or 2 watts, and its PV primer puts the theoretical efficiency limit of a silicon cell at about 32 percent, with the best commercial panels around 18 to 22 percent, partly because only light above the band gap is absorbed.

A photodiode is the same junction run as a sensor, usually with reverse bias, so the photocurrent is proportional to the light. The pixels in a CMOS image sensor are photodiodes with readout circuitry beside them.

The threshold logic carries over. Any photon whose energy falls short of the band gap passes through without freeing a carrier, which is why silicon sensors and solar cells stop responding in the near infrared and why an absorber's band gap sets the long wavelength edge of its response.

Where does the photoelectric effect show up today?

Photomultiplier tubes. A photomultiplier starts with a photocathode, a surface chosen for a low work function and good quantum efficiency. Each photoelectron is accelerated onto a chain of electrodes called dynodes, each of which releases several secondary electrons, so one photon becomes a measurable pulse. The IceCube Neutrino Observatory at the South Pole is a large scale example: its in ice detector has 5,160 digital optical modules, each with a ten inch photomultiplier tube, frozen into 86 boreholes between 1,450 and 2,450 metres deep. The tubes register faint flashes from particles produced when neutrinos interact in the ice.

Photoemission spectroscopy. Flip the equation around and it becomes a measuring tool. If you know hf precisely and measure the electron's kinetic energy, you can find how tightly the electron was bound. In X-ray photoelectron spectroscopy (XPS), the kinetic energy is roughly hf minus the binding energy minus a work function term for the spectrometer, and the binding energies are fingerprints of the elements and their chemical states. Lawrence Berkeley National Laboratory's X-Ray Data Booklet tabulates electron binding energies for the elements and notes that for metals they are given relative to the Fermi level, for gases relative to the vacuum level, and for semiconductors relative to the top of the valence band, which is the same reference level bookkeeping as the work function. Angle resolved photoemission spectroscopy (ARPES) also records the direction of each electron, mapping energy against momentum in a crystal; Berkeley Lab's Advanced Light Source runs a dedicated ARPES programme on its synchrotron beamlines.

Image sensors. As the 2009 Nobel press release puts it, CCD sensors turn light into electric signals through the photoelectric effect. CMOS sensors, now dominant in phones, use the same internal effect in photodiodes. See how camera image sensors are made for the manufacturing side.

Night vision and image intensifiers. Image intensifier tubes put a photocathode at the front: incoming light from a dim scene frees photoelectrons, which are amplified and accelerated onto a phosphor screen that glows with a brighter copy of the image. The physics at the input is the external photoelectric effect; the choice of photocathode material sets which wavelengths the device sees.

Choosing a cathode is choosing a threshold. The table below is the design rule in miniature. A detector meant to see visible light needs a low work function surface, like the alkali metals at the top of the table. A detector meant to ignore daylight and respond only to short ultraviolet wavelengths can use a surface with a high work function, so visible photons simply fall short of the barrier.

Table of work functions and threshold wavelengths

Work functions below are from the HyperPhysics table (Georgia State University), which cites the Handbook of Chemistry and Physics for some entries. Threshold frequencies and wavelengths are our calculations using the exact SI constants. Treat the work functions as typical values for clean polycrystalline surfaces; real samples differ.

HyperPhysics also gives alternative values for two alkali metals, 2.29 eV for potassium and 2.36 eV for sodium, which it attributes to polycrystalline samples. With 2.36 eV, sodium's threshold moves to 525.4 nm. Differences of that size are normal between sources.

Walkthrough: reproduce every number in Python

Everything above can be checked with a short script that uses only the Python standard library and the exact SI constants. The steps below set up an isolated environment, run the calculator, and optionally cross check against SciPy, whose documentation states that scipy.constants carries the 2022 CODATA recommended values.

Step 1: create a project folder and a virtual environment.

mkdir photoelectric && cd photoelectric
python3 -m venv .venv
source .venv/bin/activate          # on Windows: .venv\Scripts\activate
python --version

Step 2: save the calculator as photoelectric.py.

"""Photoelectric effect calculator: K_max = h*f - phi."""
import math

# Exact SI defining constants (2019 SI), as published by BIPM and NIST CODATA
H = 6.62607015e-34          # Planck constant, J s (exact)
E_CHARGE = 1.602176634e-19  # elementary charge, C (exact)
C = 299792458               # speed of light, m/s (exact)
M_E = 9.1093837139e-31      # electron mass, kg (CODATA 2022, measured)

HC_OVER_E_NM = H * C / E_CHARGE * 1e9   # 1239.84198... nm eV

def photon_energy_ev(wavelength_nm):
    return HC_OVER_E_NM / wavelength_nm

def threshold(phi_ev):
    f0 = phi_ev * E_CHARGE / H
    lam0_nm = HC_OVER_E_NM / phi_ev
    return f0, lam0_nm

def emission(wavelength_nm, phi_ev):
    e_ph = photon_energy_ev(wavelength_nm)
    k = e_ph - phi_ev
    if k <= 0:
        return e_ph, None, None, None
    v = math.sqrt(2 * k * E_CHARGE / M_E)
    return e_ph, k, k, v   # K_max in eV equals stopping potential in volts

if __name__ == "__main__":
    print(f"hc/e = {HC_OVER_E_NM:.6f} nm eV, h/e = {H/E_CHARGE:.9e} V s")
    f0, lam0 = threshold(4.3)
    print(f"Zinc: f0 = {f0:.4e} Hz, lambda0 = {lam0:.1f} nm")
    e_ph, k, v0, v = emission(253.6517, 4.3)
    print(f"253.65 nm on zinc: photon {e_ph:.3f} eV, K_max {k:.3f} eV, "
          f"V0 {v0:.3f} V, v_max {v:.3e} m/s")
    # Millikan-style straight-line fit on ideal sodium data (phi = 2.28 eV)
    lines_nm = [253.6517, 312.5668, 365.0153, 404.6563, 435.8328, 546.0735]
    pts = []
    for lam in lines_nm:
        e_ph, k, v0, _ = emission(lam, 2.28)
        f = C / (lam * 1e-9)
        print(f"{lam:9.4f} nm  f = {f/1e14:6.3f}e14 Hz  V0 = "
              + (f"{v0:.3f} V" if v0 else "no emission"))
        if v0:
            pts.append((f, v0))
    n = len(pts)
    mx = sum(p[0] for p in pts) / n
    my = sum(p[1] for p in pts) / n
    slope = sum((x - mx) * (y - my) for x, y in pts) / sum((x - mx) ** 2 for x, _ in pts)
    icpt = my - slope * mx
    print(f"slope = {slope:.6e} V s -> h = {slope*E_CHARGE:.6e} J s, "
          f"phi = {-icpt:.3f} eV, f0 = {-icpt/slope:.4e} Hz")

Step 3: run it.

python photoelectric.py

Expected output:

hc/e = 1239.841984 nm eV, h/e = 4.135667697e-15 V s
Zinc: f0 = 1.0397e+15 Hz, lambda0 = 288.3 nm
253.65 nm on zinc: photon 4.888 eV, K_max 0.588 eV, V0 0.588 V, v_max 4.548e+05 m/s
 253.6517 nm  f = 11.819e14 Hz  V0 = 2.608 V
 312.5668 nm  f =  9.591e14 Hz  V0 = 1.687 V
 365.0153 nm  f =  8.213e14 Hz  V0 = 1.117 V
 404.6563 nm  f =  7.409e14 Hz  V0 = 0.784 V
 435.8328 nm  f =  6.879e14 Hz  V0 = 0.565 V
 546.0735 nm  f =  5.490e14 Hz  V0 = no emission
slope = 4.135668e-15 V s -> h = 6.626070e-34 J s, phi = 2.280 eV, f0 = 5.5130e+14 Hz

Step 4 (optional): cross check the constants against SciPy. Installing a scientific library pulls in compiled wheels and their dependencies, so install it into the virtual environment and record exactly what you got.

python -m pip install scipy
python -m pip freeze > requirements.txt
python -c "from scipy import constants as k; print(k.h, k.e, k.c, k.h*k.c/k.e*1e9)"

The last line should print the same exact values for h, e and c and 1239.84198... for hc/e in nm eV.

Step 5 (optional): make the install reproducible. pip's secure installs guide describes hash checking mode: add a --hash entry for each requirement, then install with --require-hashes, and optionally --only-binary :all: to refuse source distributions. pip hash computes the digest of a downloaded file.

python -m pip download scipy --only-binary :all: -d wheels
python -m pip hash wheels/*.whl
# add each printed --hash=sha256:... to the matching line in requirements.txt, then:
python -m pip install --require-hashes --only-binary :all: -r requirements.txt

Hash checking is all or nothing: once one requirement carries a hash, pip requires hashes for every package, including dependencies, so pin the full set from pip freeze.

Step 6 (optional): check the project's dependencies for known vulnerabilities. If this script grows into a lab data pipeline or a dashboard, scan the folder like any other codebase:

safeguard scan --dir .

Results are written as JSON under ./output. The scan looks at declared and installed open source components; it says nothing about whether your physics is right.

Common mistakes

• Using intensity where frequency belongs. Brighter light below threshold still produces no photoelectrons. Brighter light above threshold produces more electrons with the same maximum energy.
• Forgetting "maximum". hf minus φ is the energy of the fastest electrons, those that leave from the surface without losing energy. Most photoelectrons leave slower, which is why you measure the stopping potential rather than an average speed.
• Mixing eV and joules. Subtracting a work function in eV from a photon energy in joules gives nonsense. Convert both to the same unit first; the eV route with 1239.84 is usually simpler.
• Using nanometres in E = hc/λ with SI constants. If you plug λ = 253.65 into hc/λ with h and c in SI units, you are off by a factor of a billion. Convert to metres, or use the nm eV shortcut.
• Treating the work function as exact. Surface oxides, adsorbed gas and crystal orientation shift φ by tenths of an electronvolt. Silver alone spans 4.26 to 4.74 eV in the HyperPhysics notes. Quote answers to the precision the work function supports.
• Ignoring contact potential. In a real tube the collector and photocathode are usually different metals, and their work function difference adds a built in voltage. The measured stopping potential is shifted by it, which is one reason the intercept of a Millikan plot is less reliable than its slope.
• Confusing the external and internal effects. A solar cell does not eject electrons into vacuum, and its threshold is a band gap, not a work function. The physics is related, the numbers are different.
• Calling Einstein's Nobel a relativity prize. The citation names the photoelectric effect. His Nobel lecture happened to be on relativity.
• Saying Millikan proved photons exist. Millikan confirmed Einstein's equation and measured h, while publicly doubting the light quantum hypothesis behind it, as the APS centennial article documents.
• Using pre 2019 constants and reporting more digits than they support. With h = 6.63 × 10⁻³⁴ and e = 1.60 × 10⁻¹⁹, answers are only good to two or three figures. With the exact SI values, the constants are no longer the limiting factor.
• Using the non relativistic speed formula at X-ray energies. For a 0.6 eV electron it is fine. For photoelectrons with tens of keV or more, use the relativistic expression.

FAQ

What is the photoelectric effect in simple terms?

Light knocks electrons out of a surface. Each particle of light, a photon, carries energy hf. If that energy is larger than the work function, the energy needed to free an electron from that surface, an electron can escape, carrying the leftover energy as motion. If not, nothing happens, however bright the light.

What is the photoelectric equation?

K_max = hf minus φ. K_max is the maximum kinetic energy of the emitted electrons, h is the Planck constant (exactly 6.626 070 15 × 10⁻³⁴ J s), f is the light's frequency and φ is the work function. In stopping potential form, eV₀ = hf minus φ.

How do you calculate the threshold frequency?

Divide the work function by the Planck constant: f₀ = φ/h. For zinc at 4.3 eV, f₀ = 4.3 × 1.602 176 634 × 10⁻¹⁹ J ÷ 6.626 070 15 × 10⁻³⁴ J s = 1.04 × 10¹⁵ Hz. The threshold wavelength is λ₀ = c/f₀, or 1239.84 divided by φ in eV, which gives 288.3 nm for zinc.

Why doesn't brighter light give electrons more energy?

Because energy is delivered one photon at a time, and each photon's energy depends only on its frequency. Brighter light means more photons per second, so more electrons per second, but each electron still gets the energy of one photon. This was the observation, reported by Lenard in 1902, that classical wave theory could not explain.

What is the stopping potential?

It is the reverse voltage on the collector that just stops the fastest photoelectrons, so the photocurrent drops to zero. It satisfies eV₀ = K_max. Because of how the electronvolt is defined, the stopping potential in volts equals K_max in electronvolts.

Did Einstein win the Nobel Prize for relativity?

No. The Nobel Prize in Physics 1921 was awarded to Einstein for his services to theoretical physics and especially for his discovery of the law of the photoelectric effect. The prize was reserved in 1921 and presented in 1922. His Nobel lecture in July 1923 was nonetheless about relativity.

How did Millikan measure Planck's constant?

He illuminated freshly shaved metal surfaces in vacuum with monochromatic light, measured the stopping potential at several frequencies, and plotted stopping potential against frequency. The slope of that straight line is h/e. Using his oil drop value for e, he obtained h = 6.57 × 10⁻²⁷ erg s, published in Physical Review on 1 March 1916.

Is the photoelectric effect the same as how solar panels work?

They are related, not the same. Both start with a photon giving its energy to an electron. In the photoelectric effect the electron leaves the material. In a solar cell the electron stays inside the semiconductor and is pushed by the cell's internal electric field to the contacts, so the cell produces power without any applied voltage.

What is the work function of common metals?

Typical values from the HyperPhysics table are cesium 2.1 eV, sodium 2.28 eV, potassium 2.3 eV, aluminium 4.08 eV, zinc 4.3 eV, copper 4.7 eV, gold 5.1 eV and platinum 6.35 eV. Real surfaces vary with cleanliness and crystal face.

Why did the 2019 SI change matter for photoelectric calculations?

Since 20 May 2019 the Planck constant, the elementary charge and the speed of light have exact defined values. That makes h/e, hc/e and the joule value of one electronvolt exact. Uncertainty in a photoelectric calculation now comes only from the work function and the measured wavelength.

The software around the instruments: an honest bridge

The photoelectric effect is physics, and Safeguard does not do physics. Its models are single purpose security models for software; they do not calculate photon energies, fit Millikan plots, analyse detectors or inspect hardware, and nothing in this post was computed by them. The connection is narrower and real. Spectrometer control software, detector data acquisition systems, camera firmware and the analysis scripts that turn stopping potentials into constants are software, and much of it is built from open source libraries pulled from package registries. Where Safeguard helps is there: scanning those repositories, container images and SBOM files for known vulnerable open source components, showing which projects carry a given version, and, when a fixed version exists, opening an upgrade pull request for review.

Every rebuild of a lab analysis pipeline or detector control service pulls its dependencies again, and the firmware inside cameras, data loggers and instrument controllers ships open source components of its own. The fit is indirect for the physics itself, but for the code your team writes and ships around these instruments the same dependency hygiene applies, and the section below shows how to close the gap whether or not upstream has a fix yet.

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

The first demo is a real Safeguard CLI session. It installs the CLI, opens the interactive terminal with safeguard, and asks it to fix the lodash CVE in a repository: the assistant reads the finding and the advisory, asks before each command, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow applies to a Python analysis project like the walkthrough above, where the vulnerable component would be a pinned library rather than lodash.

Click through it at your own pace, or open it full screen.

The second demo sets up the local runner on your own machine, approves it with its device code, and then scans from the terminal. For a lab whose instrument control code lives on machines that should not send source elsewhere, this shows how scanning can run where the code already is.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. For an engineer already working with Claude on a detector dashboard or data pipeline, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• NobelPrize.org, Albert Einstein, Facts (Nobel Prize in Physics 1921)
• NobelPrize.org, The Nobel Prize in Physics 1921, summary and reservation note
• NobelPrize.org, Nobel Prize in Physics 1921, award ceremony speech
• NobelPrize.org, Albert Einstein, Nobel Lecture (Gothenburg, 11 July 1923)
• NobelPrize.org, Robert A. Millikan, Facts (Nobel Prize in Physics 1923)
• NobelPrize.org, Nobel Prize in Physics 1923, award ceremony speech
• NobelPrize.org, Robert A. Millikan, Nobel Lecture (23 May 1924)
• NobelPrize.org, Philipp Lenard, Facts (Nobel Prize in Physics 1905)
• NobelPrize.org, Nobel Prize in Physics 2009, press release (CCD sensor)
• American Physical Society, R. A. Millikan, A Direct Photoelectric Determination of Planck's h, Phys. Rev. 7, 355 (1916)
• American Physical Society, Physical Review Focus, Centennial Focus: Millikan's Measurement of Planck's Constant (1999)
• Internet Archive, Annalen der Physik und Chemie 1887, Vol 31 Iss 8 (Hertz, p. 983)
• Internet Archive, Annalen der Physik 1905, Vol 17 table of contents (Einstein, p. 132)
• NIST CODATA, Planck constant
• NIST CODATA, Planck constant in eV/Hz
• NIST CODATA, elementary charge
• NIST CODATA, speed of light in vacuum
• NIST CODATA, electron mass
• NIST, SI Redefinition
• NIST, Meet the Constants
• BIPM, SI defining constants
• BIPM, The International System of Units (SI Brochure)
• NIST, Handbook of Basic Atomic Spectroscopic Data, strong lines of neutral and ionised mercury
• HyperPhysics (Georgia State University), Work Functions for Photoelectric Effect
• HyperPhysics (Georgia State University), Photoelectric effect, early data and Planck hypothesis
• Lawrence Berkeley National Laboratory, X-Ray Data Booklet, Section 1.1 Electron Binding Energies
• Advanced Light Source, Berkeley Lab, angle resolved photoemission spectroscopy (ARPES)
• IceCube Neutrino Observatory, IceCube detector description
• US Department of Energy, Solar Photovoltaic Cell Basics
• US Department of Energy, Solar Photovoltaic Technology Basics
• US Department of Energy, PV Cells 101: A Primer on the Solar Photovoltaic Cell
• US Department of Energy, How Does Solar Energy Work?
• SciPy documentation, Constants (scipy.constants)
• pip documentation, Secure installs

Last reviewed: 24 October 2026.