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Light quanta · Section 8
Brighter light, more electrons.Higher frequency, faster ones.
Why does increasing light intensity release more electrons without increasing their individual energy, while increasing frequency increases electron energy without requiring higher intensity?
The photoelectric apparatus
Photoelectric apparatus laboratory
Static worked example
Model note
Incident radiant power (incidentPower), Frequency (frequency), Work function (workFunction), Quantum efficiency (quantumEfficiency) and Collector potential (collectorPotential): Host calculation, by the site's own reference code. Computed in lq08.acceptedInputs.
Quantum energy (quantumEnergy): Host calculation, by the site's own reference code. Computed in photoelectric.quantumEnergy.
Threshold frequency (thresholdFrequency): Host calculation, by the site's own reference code. Computed in photoelectric.thresholdFrequency.
Maximum photoelectron kinetic energy (maxKineticEnergy): Host calculation, by the site's own reference code. Computed in photoelectric.kMax.
Stopping potential magnitude (stoppingPotentialMagnitude): Host calculation, by the site's own reference code. Computed in photoelectric.stoppingPotentialMagnitude.
Incident quantum rate (quantumRate): Host calculation, by the site's own reference code. Computed in photoelectric.quantumRate.
Photoelectron emission rate (emissionRate): Host calculation, by the site's own reference code. Computed in photoelectric.emissionRate.
Photocurrent (photocurrent): Host calculation, by the site's own reference code. Computed in photoelectric.photocurrent.
Computed from the settings last applied (settings revision 1, result 1).
Not modeled: Real-material electron energy distributions and yields; Contact potentials and surface states; Space charge; Reflection losses; Emission angles; Multi-photon or thermionic emission; The timing of individual emissions; Energy transfer models beyond the declared complete or partial cases; Any claim that the moving marks depict photons.
Predict before the numbers
Make the lamp twice as bright without changing its frequency. What happens to the energy of the fastest electrons?
Predict before the numbers
Raise the frequency while keeping the lamp's power the same. What happens to the number of quanta arriving each second?
Predict before the numbers
Two different metals are lit by the same lamp. Plotted against frequency, are their stopping-potential lines parallel, crossing, or identical?
The result appears when you choose, say you have one in mind, or skip.
600.0 THz: one quantum carries hν = 2.48 eV
1.00 mW
2.515 × 1015 quanta arrive each second.
Experiment settings work function, quantum efficiency, collector potential, Millikan’s 1916 data
2.20 eV, a hypothetical metal unless a preset names one
Worked example: a quantum of 600 THz light carries 2.48 eV, more than the 2.2 eV work function, so the fastest electrons leave with 0.28 eV and a stopping potential of 0.28 V holds them back.
In the model the slope is h/e = 4.136 × 10−15 V·s for every metal.
Millikan, 1916: sodium, on his own scale
Later evidence, published 1916. Not on the 1904 shelf.
His Fig. 6 plots, for each mercury line, the potential on the sodium at which its photocurrent stops, before the contact potential between sodium and collector is taken out. So the points sit on his scale, not the model's, and what compares is the slope.
Line through the 5 points he says fixed the slope (filled): 4.07 × 10−15 V·s ± 8.6 × 10−17, fitted here to his points as read from the figure.
Slope from Fig. 6, as printed: 4.124 × 10−15 V·s
Slope with Table II's nine determinations, as printed: 4.128 × 10−15 V·s
h/e from this laboratory's constants: 4.1357 × 10−15 V·s
His points
Frequency (THz)
Potential (V)
Fixed the slope
547.4
−2.05
yes
692.5
−1.492
yes
741.2
−1.3
yes
821.2
−0.919
yes
961.4
−0.38
yes
1182
+0.52
no
R. A. Millikan, A Direct Photoelectric Determination of Planck's "h", Physical Review (2), 7 (3), 355–388 (1916). Fig. 6.
Current against collector potential
Saturation current Isat = 40.30 μA
Values at these settings
Quantity
Value
Energy of one quantum, hν
2.481 eV
Threshold frequency, Φ/h
532 THz
Largest electron energy, hν − Φ
0.2814 eV
Stopping potential
0.2814 V
Quanta arriving each second
2.515 × 1015 per second
Electrons freed each second
2.515 × 1014 per second
Current at this collector potential
40.3 μA
Light shining on a metal knocks electrons out of it. In this model, brighter light releases more of them each second but none faster, while light of a higher frequency makes the fastest ones faster, as Einstein expected if light gives up its energy in separate quanta.
Section 8 applies the light quantum to cathode rays produced by light. In the simplest picture, which Einstein says he will assume, one quantum gives its whole energy hν to one electron; the electron spends a work Φ, characteristic of the body, in leaving it, so the fastest electrons come out with kinetic energy hν − Φ. A body charged positive just enough to keep them all in, to the potential Vs, satisfies eVs = hν − Φ. Two consequences follow. Plotted against ν, Vs is a straight line whose slope, h/e, does not depend on the substance. And if each quantum acts independently of the rest, the intensity of the light changes how many electrons leave each second, not how fast they go. The defaults, 600 THz light at 1 mW on a hypothetical surface with Φ = 2.2 eV, give quanta of 2.48 eV, a threshold at 532 THz and a stopping potential of 0.281 V; with one quantum in ten releasing an electron, 2.52 × 1014 electrons leave each second, a current of 40.3 μA. Doubling the power doubles the current and leaves 0.281 V unchanged.
Start with one quantum. Its energy is hν, Planck's constant times the frequency: h = 6.626 × 10−34 J s and ν = 6.00 × 1014 Hz give hν = 3.976 × 10−19 J. An electronvolt, the energy an electron gains in falling through one volt, is 1.602 × 10−19 J, so hν = 3.976/1.602 = 2.48 eV. To leave the metal an electron must do the work Φ, here 2.2 eV. If the quantum gives all its energy to one electron at the surface, the most that electron can keep is 2.481 − 2.2 = 0.281 eV. To stop it, charge the metal positive: an electron climbing back against a potential V loses eV of energy, so it turns back once eV equals its kinetic energy, and the fastest one needs Vs = 0.281 V. That is the stopping potential. A quantum that carries less than Φ releases nothing, so the light must have hν at least Φ, or ν at least ν0 = Φ/h = (2.2 × 1.602 × 10−19)/(6.626 × 10−34) = 5.32 × 1014 Hz, which is 532 THz. Now the intensity. A power of 1 mW delivers 10−3 J each second; shared into quanta of 3.976 × 10−19 J, that is 10−3/(3.976 × 10−19) = 2.52 × 1015 quanta each second. If one in ten releases an electron, 2.52 × 1014 electrons leave each second, and since each carries 1.602 × 10−19 C the current is 2.52 × 1014 × 1.602 × 10−19 = 4.03 × 10−5 A, or 40.3 μA. Double the power to 2 mW and there are twice as many quanta, so twice the current, 80.6 μA; each quantum still carries 2.48 eV, so the fastest electron still needs 0.281 V to stop. Einstein's own check sets the work to nothing, Φ = 0, and takes ν = 1.03 × 1015 Hz, where the solar spectrum ends in the ultraviolet. Then eVs = hν gives 4.26 V with today's constants. He printed 4.3 V and said it agrees in order of magnitude with Lenard's results.
Modern qualifications
Einstein wrote the quantum's energy as (R/N)βν, with R the gas constant, N the number of molecules in a gram-molecule and β the constant in Wien's exponent, so (R/N)β is Planck's h. He wrote Π for the potential, ε for the electron's charge and P for the work, and also gave the law per gram-equivalent of charge, ΠE = Rβν − P′. He called complete transfer to one electron the simplest picture, and allowed that an electron might take up only part of a quantum, in which case ΠE + P′ ≤ Rβν. In 1905 the prediction was untested: Lenard had reported in 1902 that the electrons' speed does not depend on the intensity of the light, but nobody had yet measured the straight line Einstein predicted. Millikan did so in 1916, finding a slope that gave Planck's h to within about half a percent, while calling the theory by which Einstein had reached the equation untenable. The word photon is Lewis's, from 1926.
Show the code
The site’s own reference code for this law, in TypeScript. It computes the numbers above when the page runs it, and it is what the page uses wherever a FrankenSim result is not available.
stoppingPotentialMagnitude · src/physics/reference/photoelectric.ts · this site’s source, as built · sha256:d3b770e4a65e0a36833d8dd82384f33a3b6e6909190c9b094ceca3043aafc29a
This function computes the listed outputs when it runs.
In words
V_s = K_max / e = (h nu - Phi) / e. The physics is in kMax, which this calls; what this function does is ask for the maximum kinetic energy in joules and divide by the elementary charge to put it in volts. Measured at 10^15 Hz against a 2.0 eV work function: 2.135667696923859 V.
Implementation
/**
* Stopping potential magnitude: V_s = K_max / e = (h * nu - Phi) / e (in Volts).
*
* Epistemic rule:
* When nu < nu_0, stopping potential is `not-applicable` with reason
* "no emitted electron in this model".
* Under partial transfer, returns `underdetermined` with upperBound = (h * nu - Phi) / e.
*/exportfunctionstoppingPotentialMagnitude(nu:number,workFunctionJoules:number,set?:ConstantSet,transferModelOrOptions:|EnergyTransferMode|Readonly<{transferModel?:EnergyTransferMode}>="complete",):PhotoelectricResult<number>{consttransferModel:EnergyTransferMode=typeoftransferModelOrOptions==="object"&&transferModelOrOptions!==null?(transferModelOrOptions.transferModel??"complete"):transferModelOrOptions;constkResult=kMax(nu,workFunctionJoules,set,transferModel);if(kResult.status==="underdetermined"){conste=getElementaryCharge(set);constextra:{upperBound?:number;unit?:string;citation?:string;reason?:string;bounds?:readonly[number,number];quantityId?:string;}={unit:"V",quantityId:"stoppingPotentialMagnitude",};if(kResult.upperBound!==undefined)extra.upperBound=kResult.upperBound/e;if(kResult.citation!==undefined)extra.citation=kResult.citation;if(kResult.reason!==undefined)extra.reason=kResult.reason;if(kResult.bounds!==undefined)extra.bounds=[kResult.bounds[0]/e,kResult.bounds[1]/e];returnunderdetermined(kResult.compatibleFamily,kResult.neededInformation,extra);}if(kResult.status!=="value")returnkResult;conste=getElementaryCharge(set);returnok(kResult.value/e,"V",{...(kResult.note?{note:kResult.note}:{}),quantityId:"stoppingPotentialMagnitude",});}
Einstein’s 1905 §8 check, by order of magnitude
What was neglected: Einstein sets P' = 0 as a deliberate neglect of escape work for order-of-magnitude comparison against Lenard's spark-potential observations, not as a physical prediction for a named metal.
What it is not: This is not a prediction for any named metal; any real substance has P' > 0, so its stopping potential at this frequency is lower by exactly the amount the work function contributes.
As printed: Π = Rβν / E = 4.34 V, “ca. 4,3 Volt”. Slope 4.212 × 10−15 V·s, against a modern h/e of 4.136 × 10−15 V·s.
At these settings: ν = 600.0 THz gives hν = 2.481 eV; with the hypothetical Φ = 2.20 eV the stopping potential is 0.281 V.
What this model leaves out
It follows Einstein’s 1905 rule that one absorbed quantum gives its energy to one electron, which spends Φ escaping. It does not model:
Real-material electron energy distributions and yields
Contact potentials and surface states
Space charge
Reflection losses
Emission angles
Multi-photon or thermionic emission
The timing of individual emissions
Energy transfer models beyond the declared complete or partial cases
Any claim that the moving marks depict photons
Compare two setups side by side in your reading. Each laboratory has its own settings, stepwise state and accepted results.
The physical argument
The single-quantum energy conservation law
In §8 of his 1905 paper, Einstein applies the light-quantum hypothesis to the generation of cathode rays by light (the photoelectric effect). If monochromatic light consists of energy quanta of magnitude NRβν=hν, the simplest picture, which Einstein says he will assume, is that an absorbed quantum gives its entire energy to a single electron in the body. For one electron, of charge ε, the body's stopping potential Π satisfies:
Einstein also writes the same law for a gram-equivalent of charge, ΠE=Rβν−P′, which is the one-electron law multiplied by N. In modern notation, writing Π as the stopping potential Vs, ε as the elementary charge e, and P as the surface escape work Φ:
The electrons' speed does not depend on the intensity. If each quantum gives up its energy independently of the others, more intense light brings more quanta each second (N˙=Plight/(hν)), so the number of electrons leaving is proportional to the intensity, while the spread of their speeds stays the same. Lenard had reported in 1902 that the speed did not depend on the intensity.
The potential is a straight line in the frequency. Plotted against ν, the potential must be a straight line whose slope does not depend on the substance. Below ν0=Φ/h no single quantum carries enough energy to release an electron. Einstein expected these rules to have limits, as he expected for Stokes's rule in §7: where the light is so intense that energy from several quanta can combine, or where it is far from the range of Wien's law.
Einstein's 1905 historical check
In 1905 there were few measurements to compare with. Philipp Lenard had measured, in 1902, the potentials that bodies illuminated by arc and spark light reach. Einstein took the ultraviolet end of the solar spectrum, ν=1.03×1015s−1 (a wavelength near 290 nm), neglected the escape work, and found:
Vs≈ehν≈4.3V
He wrote that this agrees in order of magnitude with Lenard's results. It is a check of scale, not a measurement of the law.
The slope, measured in 1916
Robert Millikan tested the straight line on alkali metals cut clean in a vacuum (sodium, potassium, lithium). His 1916 lines had the slope dνdVs=eh within his errors, giving Planck's h to about half a percent, and differed from metal to metal only in where they crossed zero, at ν0. Millikan still called the theory by which Einstein had reached the equation untenable.
Epistemic boundary: deductive consequences vs. empirical proof
A simulator programmed with an energy threshold does not prove that nature has a threshold; it demonstrates the deductive consequences of single-quantum energy exchange and surface escape work (Eq=hν, W=Φ). Independent experiments (such as Millikan’s 1916 measurements and later single-photon anti-bunching measurements) test whether those assumptions describe the physical world.
Below ν0 there is no stopping potential to report: no electron leaves, so there is nothing to stop, and the instrument says so rather than showing zero. Millikan's 1916 result is later evidence, not a premise of the 1905 argument.
Einstein's photoelectric law was an established experimental fact in 1905.
In 1905 the law was a prediction. The paper checks only an order of magnitude, and states what must hold if the formula is correct. The measurement that tested the straight line came in 1916.