Work Function Calculator
Enter photon wavelength λ and maximum photoelectron kinetic energy KE to compute work function φ=hc/λ−KE and threshold frequency ν0=φ/h. Na λ=400nm, KE=0.82eV → φ≈2.28eV, λ₀≈544nm; W λ=200nm, KE=1.7eV → φ≈4.5eV.
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At a glance:The work function (φ, symbol Φ) is the minimum energy (eV) for an electron to escape a metal into vacuum, equal to the gap between the vacuum level and the Fermi level. Photoelectric effect (Einstein, 1905): when a metal surface is illuminated, electrons absorb photon energy hν and escape; energy conservation gives hν=φ+KE_max, so φ=hν−KE=hc/λ−KE. When hν<φ no photoelectric effect occurs, corresponding to threshold frequency ν0=φ/h and threshold wavelength λ0=hc/φ=c/ν0. Three hallmarks of the photoelectric effect: (1) a threshold frequency exists — below ν0 no emission regardless of intensity; (2) max kinetic energy is proportional to light frequency and independent of intensity; (3) immediate emission (<10⁻⁹ s), no accumulation time. These cannot be explained by wave theory and are key evidence for the photon hypothesis. Metal φ ranges 2-6 eV: alkali metals lowest (Cs 2.1, K 2.3, Na 2.28), transition metals higher (Zn 4.3, Ag 4.26, Cu 4.65, Fe 4.5, W 4.5, Pt 5.65, Ni 5.01). The work function depends on surface (different crystal faces), coverage (Cs lowers W φ to 1.5 eV) and temperature (small variations). Classic example: Na λ=400nm, KE=0.82 eV → hν=hc/λ=1240/400=3.10 eV, φ=3.10−0.82=2.28 eV, λ0=1240/2.28=544 nm (yellow-green), ν0=φ·e/h=2.28×1.602e-19/6.626e-34=5.51×10¹⁴ Hz; W λ=200nm, KE=1.7 eV → hν=1240/200=6.20 eV, φ=6.20−1.70=4.50 eV, λ0=1240/4.5=276 nm (UV). History: Hertz discovered the photoelectric effect in 1887; Lenard observed the threshold frequency in 1902; Einstein proposed the light-quantum explanation in 1905 (Nobel 1921); Millikan verified precisely in 1916. Applications: (1) photomultiplier tubes — detect weak light; (2) photovoltaic cells — solar power; (3) night-vision devices — infrared photocathodes; (4) electron guns — electron microscopy, CRT; (5) surface analysis — photoelectron spectroscopy (XPS, UPS).
Formula
Work function: φ = hν − KE = hc/λ − KE
Threshold frequency: ν0 = φ / h
Threshold wavelength: λ0 = c / ν0 = hc / φ
Photon energy: E = hν = hc/λ (eV·nm=1240/λ)
Stopping voltage: KE = e·V0
$$\varphi = h\nu - KE_{\max} = \frac{hc}{\lambda} - KE_{\max}, \quad \nu_0 = \frac{\varphi}{h}, \quad \lambda_0 = \frac{hc}{\varphi}$$How to Use
- Enter incident wavelength λ (nm) and max photoelectron kinetic energy KE (eV).
- The tool computes φ=hc/λ−KE, ν0=φ/h, λ0=hc/φ and KE (J).
- Common: Na λ=400nm, KE=0.82 → φ=2.28 eV, λ0=544nm; W λ=200nm, KE=1.7 → φ=4.5 eV.
Work Functions and Photoelectric Thresholds of Common Metals
| Metal | φ (eV) | φ (J) | ν0 (Hz) | λ0 (nm) | Spectrum |
|---|---|---|---|---|---|
| Cesium | 2.10 | 3.364e-19 | 5.08e14 | 590 | Orange-yellow |
| Potassium | 2.30 | 3.685e-19 | 5.56e14 | 539 | Yellow-green |
| Sodium | 2.28 | 3.653e-19 | 5.51e14 | 544 | Yellow-green |
| Calcium | 2.87 | 4.598e-19 | 6.94e14 | 432 | Blue-violet |
| Zinc | 4.30 | 6.889e-19 | 1.04e15 | 288 | UV |
| Silver | 4.26 | 6.825e-19 | 1.03e15 | 291 | UV |
| Copper | 4.65 | 7.450e-19 | 1.12e15 | 267 | UV |
| Iron | 4.50 | 7.209e-19 | 1.09e15 | 276 | UV |
| Tungsten | 4.50 | 7.209e-19 | 1.09e15 | 276 | UV |
| Nickel | 5.01 | 8.026e-19 | 1.21e15 | 248 | UV |
| Platinum | 5.65 | 9.052e-19 | 1.37e15 | 219 | UV |
hc≈1240 eV·nm, h=6.626×10⁻³⁴ J·s, c=2.998×10⁸ m/s, e=1.602×10⁻¹⁹ C. φ depends on surface state and crystal face; literature values vary by ±0.1 eV.
Case Studies
Photomultiplier Sensitive Material Selection
Photomultiplier tubes (PMT) must detect photons from visible to UV; the photocathode material affects quantum efficiency. Bialkali Sb-K-Cs has φ≈2.0 eV, λ0=620nm, covering visible to 400nm.
Ag-Sb-Cs cathodes with φ≈1.8 eV extend to near-infrared 700nm but have larger dark current, hurting signal-to-noise.
Medical PET scanners use PMTs to detect scintillation light (blue 420nm) from 511keV gamma excitation, with ~25% quantum efficiency; single-photon counting capability is key to CT imaging.
Electron-Beam Lithography and Tungsten-Filament Electron Guns
SEM/TEM electron guns use tungsten wire φ=4.5 eV, heated to 2500K for thermionic emission, accelerated at 1-200 keV to form the beam.
Field-emission guns (FEG) use a single-crystal tungsten tip with enhanced electric field to lower the barrier, improving current density 1000×, reaching 0.1nm (atomic) resolution.
A tungsten-filament gun costs ~HK$1000 with ~100 h life; an FEG costs ~HK$500k with ~5000 h life. Hong Kong University's electron microscopy center uses FEGs for semiconductor failure analysis.
FAQ
Why does a threshold frequency exist?
In the photoelectric effect an electron absorbs one photon of energy hν at a time. If hν is less than the work function φ, the electron cannot gain enough energy to escape the metal, and no photoelectrons are emitted regardless of intensity. Intensity only determines the number of photons (photocurrent strength), not the energy of a single photon. The threshold frequency ν0=φ/h is the frequency at which photon energy just equals the work function.
Why is photoelectron kinetic energy independent of intensity?
The photoelectric effect is a single-photon process: one electron absorbs one photon. Increasing intensity only increases the number of photons (thus the photocurrent), but the single-photon energy hν is unchanged, so the energy gained by the electron is unchanged. Max kinetic energy KE=hν−φ depends only on light frequency and material. This contradicts the wave-theory prediction that 'kinetic energy should increase with intensity' and is key evidence for Einstein's light-quantum hypothesis.
How is a metal's work function measured?
(1) Photoelectric method — measure the stopping voltage V0 at different wavelengths, KE=eV0, plot hν vs V0: slope 1, intercept −φ/e; (2) Thermionic-emission method — measure current density at high temperature, invert the Richardson equation; (3) Contact-potential method — measure the contact potential difference between two metals; (4) Field-emission method — measure emission current under strong field; (5) Photoelectron spectroscopy (XPS, UPS) — measure the kinetic-energy distribution directly to obtain φ. Most common are the photoelectric method and UPS.
Why is tungsten used for filaments despite its high work function?
Tungsten φ=4.5 eV is high, but its melting point is 3422°C, with slow evaporation at high temperature and excellent mechanical strength. Heated to 2500K, thermionic emission density is sufficient (J=A·T²·exp(−φ/kT)≈6369 A/m²). With thorium treatment (a thorium monolayer lowers the effective φ to 2.6 eV), emission improves 1000×. Tungsten wire is cheap and easy to process, still the first choice for electron guns and X-ray tube filaments.
What is the relation between work function and photovoltaic cells?
Photovoltaic cells use semiconductor PN junctions rather than the metal photoelectric effect, but the work function determines electrode contact properties. The work-function difference at a metal-semiconductor contact forms a Schottky barrier, affecting carrier injection efficiency. The transparent electrode ITO (φ≈4.7 eV) forms an ohmic contact with silicon needing doping adjustment. Perovskite solar cells must match electrode φ to maximize open-circuit voltage, with efficiency exceeding 25%.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.