Count energy one photon at a time

A photon — open definition carries energy E = hf, where h is Planck’s constant and f is frequency. In vacuum this can also be written E = hc/λ, with wavelength λ and light speed c. Higher frequency means more energy per photon; longer wavelength means less. Increasing the number of same-frequency photons increases the total transferred energy without changing the energy of each one.

An electron-volt — open definition is a unit of energy, not a unit of electric potential. It is the energy associated with one elementary charge moving through one volt of potential difference. Spectroscopy often uses it because individual atomic interactions involve energies much smaller than a joule. Carefully distinguishing eV from V prevents a particle-energy label from being mistaken for a detector’s supply voltage.

For an ideal isolated atom, absorption between two specified allowed bound states requires a photon energy matching the energy gap, within the physical line width. An energetic enough photon can also ionize an atom, with energy beyond the threshold contributing to the outgoing particle’s motion and recoil. The details depend on the target and the process, rather than on photon energy alone.

The photoelectric effect supplies a related example: in the simple single-photon model, an electron can escape a material only if the photon supplies at least its work function — open definition. Raising intensity below that threshold does not give each photon a higher frequency. At high intensities other processes can matter, so the single-photon statement belongs to its stated experimental regime.

Imagine allowed atomic energy levels at 0, 2 and 5 eV relative to a chosen ground-state zero. A transition from 5 to 2 eV emits a 3 eV photon in this simplified model. A transition from 2 to 0 emits 2 eV. The arrows describe energy changes, not distances. A two-step cascade can carry the same 5 eV total as a direct transition but divide it among different photons.

Now illuminate a hypothetical material with a 2 eV work function using 3 eV photons. In the ideal photoelectric account, the maximum emitted-electron kinetic energy is 1 eV. Sending twice as many such photons may increase emitted-electron numbers under suitable conditions; it does not raise that maximum by changing the energy per photon. Losses and detector efficiency affect actual counts.

Read an energy-level difference

A detector may count incoming events, estimate deposited energy, or record a signal integrated over a wavelength band. Those are different measurements. Two exposures with equal counts can represent different received energies if their photon-energy distributions differ. Two instruments looking at the same source can also have different counts because their collecting areas and efficiencies differ.

NuSTAR’s role as an X-ray observatory gives these distinctions a practical setting. Its hardware image cannot demonstrate the spectrum of any particular object. To compare a physical emission model with a measured spectrum, include detector response, background and uncertainty. The useful question is not simply whether photons were detected, but which measured distribution distinguishes the competing explanations.