X-Rays

Walk into the radiology department at AIIMS Delhi on any weekday. The technologist slides a film cassette behind a patient's hand, presses a button for a fraction of a second, and out comes an image that a physician can read — black where soft tissue let the beam through, white where bone stopped it. What lit that film is a beam of photons whose wavelength is about ten thousand times shorter than visible light. Each photon carries roughly ten thousand times the energy of a photon of yellow light. That is why it can pass through three centimetres of flesh without being absorbed, and why it gets stopped by a millimetre of calcified bone: energy enough to interact with inner electrons, not low enough to be scattered by outer ones.

These photons did not exist five minutes before the exposure. They were made, on demand, inside a sealed glass tube the size of a mango. A hot filament in one end of the tube boils off electrons. A potential difference of 50,000 volts (or more) between the filament and a chunk of tungsten at the other end accelerates those electrons up to about 40 per cent of the speed of light. When an electron slams into the tungsten, most of its kinetic energy becomes heat in the metal. But a small fraction — a percent or two — gets radiated away as a photon. That photon is an X-ray.

The spectrum of X-rays coming out of such a tube has two components layered on top of each other, and they have completely different origins. One is a smooth, broad hump that depends only on the tube voltage — push more energy in and the hump extends to shorter wavelengths. The other is a set of sharp, narrow spikes whose wavelengths depend only on which element the target is made of — switch from tungsten to molybdenum and the spikes jump to new wavelengths. In this article you will derive where each component comes from, from scratch.

The X-ray tube — the apparatus that makes the photons

Before the physics, the hardware. An X-ray tube is a two-terminal vacuum tube:

The one number that matters for the photon energies is the accelerating voltage V. An electron crossing a potential difference V gains kinetic energy

Why: work done by the electric field on charge e moving through potential difference V is eV. All of that work becomes kinetic energy of the electron (the tube is evacuated, so nothing slows it down en route).

and slams into the target with that energy. At V = 50 kV, K = 50 keV = 8.01\times10^{-15} J, about 6\times10^4 times a visible-light photon's energy. The photon that comes out can carry at most this much energy — the electron has nothing more to give — and that is the entire content of the Duane–Hunt cutoff that you will derive next.

The continuous spectrum — bremsstrahlung and the Duane–Hunt cutoff

Most of the X-ray spectrum is a smooth, broad continuum. The physics behind it has a German name — bremsstrahlung — that literally means "braking radiation." Here is what happens:

An electron moving at high speed enters the tungsten. It does not hit the nucleus directly — the nucleus is 10^5 times smaller than the atom. Instead, it passes close to a tungsten nucleus and gets deflected by the Coulomb force. During the deflection the electron decelerates (and accelerates transversely), and any accelerated charge radiates electromagnetic waves (a fact you saw in the photoelectric effect article — it is the same radiation-by-acceleration story as in a radio antenna, just at a million times higher frequency).

The electron can lose anywhere from zero to all of its kinetic energy in a single encounter. If it barely grazes the nucleus, it radiates a low-energy photon. If it brakes hard, it radiates a high-energy photon. A typical electron makes many such encounters before coming to rest, radiating a photon of some energy at each one. The cumulative result is photons of every energy up to — but not beyond — the electron's initial eV.

The upper bound on photon energy is absolute. An electron whose total kinetic energy is eV cannot radiate a photon carrying more than eV — that would violate conservation of energy. Nothing else is conserved strictly in a single encounter (the struck atom can recoil, electrons can lose any fraction of their energy, etc.), so photons at all lower energies are possible. The maximum photon energy is therefore

Step 1. Relate photon energy to wavelength using E = hf and f = c/\lambda.

Why: shorter wavelength means higher frequency and higher energy. The maximum-energy photon is the minimum-wavelength photon.

Step 2. Set E_\text{max} = eV and solve for \lambda_\text{min}.

Why: the electron's entire kinetic energy becomes one photon in the extreme case. This happens with vanishingly small probability, but the bremsstrahlung continuum runs right up to (and stops at) this wavelength. Beyond it, the spectrum is cleanly zero — you cannot extract a photon carrying more energy than you put into an electron.

Step 3. Plug in numbers for V = 50 kV. Use the handy combination hc = 1240 eV·nm.

Why: the hc = 1240 eV·nm shortcut (seen already in photons) collapses the calculation to one division. At 50 kV the shortest wavelength is 24.8 picometres — about 200 times shorter than the smallest visible-light wavelength. That is why X-rays can resolve features at the scale of the distances between atoms in a crystal.

The relation \lambda_\text{min} = hc/(eV) is called the Duane–Hunt limit or the Duane–Hunt law, after the 1915 measurement that confirmed it. It is the cleanest quantitative test of the photon picture of light outside the photoelectric effect. At a given tube voltage V, no photon emerges at a wavelength shorter than hc/(eV). Measure \lambda_\text{min} by dispersing the X-ray beam with a crystal, and you get a direct determination of Planck's constant — the ratio eV/\lambda_\text{min} = hc is a quantity made out of measurable tube parameters and the resulting cutoff wavelength, with h as the only unknown.

The characteristic spectrum — why there are sharp lines on top

Measure the actual X-ray spectrum carefully and you find the smooth bremsstrahlung hump is decorated with a few sharp, tall spikes. A typical tungsten spectrum has a spike at 20.9 pm (the Kα line) and another at 18.4 pm (Kβ). Switch the target to copper and the spikes jump to 154 pm and 139 pm. Molybdenum: 71 pm and 63 pm. Same shape each time, wavelengths different.

What is producing these lines? The answer is the same Bohr-style energy-level physics that produced the hydrogen spectrum, except now the atom is a heavy element with 29 or 42 or 74 electrons, and the transitions happen deep in the inner shells rather than in the outer ones.

Here is the two-step process:

1. Knocking out an inner electron. An incoming 50 keV electron has enormous energy compared to the outer-shell electrons of a tungsten atom (whose binding energies are a few eV). It can reach right down and eject a K-shell electron (binding energy about 70 keV for tungsten — but this is slightly less than the 50 keV... see below). For a lighter element like copper (K binding 8.98 keV), a 50 keV electron easily knocks out a K electron.

2. Filling the vacancy. The hole in the K-shell is filled by an electron dropping from a higher shell (L, M, N, ...). When it drops, the atom emits a single photon whose energy equals the difference in binding energies:

Why: inner electrons feel almost the full nuclear charge because they are closer to the nucleus than outer electrons. When an L-shell electron drops into a K-shell vacancy, the energy of the photon released is tens of keV — deep X-ray territory. The characteristic lines are exactly the hydrogen-like Bohr transitions, scaled up by the near-full nuclear charge an inner electron sees.

This gives you a naming convention straight out of spectroscopy notation:

• Kα — L-shell electron fills a K-shell hole (n = 2 \to n = 1)
• Kβ — M-shell electron fills a K-shell hole (n = 3 \to n = 1)
• Lα — M-shell electron fills an L-shell hole (n = 3 \to n = 2)
• and so on

The Greek letter indicates how far the electron fell (α for one level, β for two, γ for three). The letter in front (K, L, M) indicates which shell the final state is in.

Moseley's law — why \sqrt{f} is linear in Z

In 1913, the same year Bohr published his hydrogen atom, the Manchester experimentalist Henry Moseley measured characteristic X-ray wavelengths for every element he could get his hands on — from aluminium to gold, crossing most of the periodic table. When he plotted \sqrt{f_{K\alpha}} versus Z (atomic number), he got a straight line:

with b \approx 1 (so Z - b is almost, but not quite, the atomic number itself). This is Moseley's law, and it is derivable from Bohr's model with one small modification.

Step 1. For a hydrogen-like ion of nuclear charge Ze, the Bohr energy levels are

Why: you derived this in the Bohr's model article. Replacing e^2 with Ze^2 in the Coulomb attraction scales the energy by Z^2.

Step 2. A tungsten K-shell electron does not quite see the full nuclear charge Z e. There is one other K-shell electron sitting at roughly the same distance from the nucleus, and on average it screens one unit of the nuclear charge. The electron of interest effectively experiences a reduced charge (Z-1)e. This inner screening is the small correction Moseley's b \approx 1 captures.

Why: Pauli exclusion puts two electrons (opposite spins) into the K shell. When one is present as a spectator, it reduces the effective nuclear charge seen by its partner by about one unit. This is a first-order approximation; the exact screening is slightly less than one because the spectator electron is not a rigid shell.

Step 3. For the Kα transition (n_i = 2 \to n_f = 1) the photon energy is the difference:

Why: this is the Bohr transition formula applied to inner electrons, with the screening correction. Because outer electrons are far away and are shielded by inner electrons, they make almost no difference to the K-shell energetics — which is why the characteristic X-ray wavelength depends only on Z and is almost completely insensitive to the chemical state of the atom.

Step 4. Solve for \sqrt{f_{K\alpha}}.

Why: the square root turns the (Z-1)^2 dependence into a linear one. The constant a = \sqrt{10.2\ \text{eV}/h} evaluates to 4.96\times 10^7 Hz^{1/2}.

Compare to Moseley's experimental form \sqrt{f} = a(Z - b): you get b = 1 exactly, from the one-electron screening assumption, and a = 4.96\times 10^7 Hz^{1/2} from the Rydberg energy. Both match Moseley's measurements almost perfectly. (The real b for Kα lines is closer to 1.1 across the periodic table — a 10% correction that comes from finer details of the screening.)

Moseley's result was more than a pretty formula. It gave atomic number a physical meaning — before 1913, elements were ordered in the periodic table by atomic weight, which occasionally put elements out of chemical order (cobalt is slightly heavier than nickel but chemically comes before it). Moseley's Z is the charge on the nucleus, full stop. It is an integer, and every integer from 1 upward corresponds to exactly one element. Gaps in his plot identified elements that had not yet been discovered (promethium, rhenium). Characteristic X-rays are not just a physics result; they remade the periodic table.

Explore the spectrum — the tube voltage in your hands

Drag the slider below from 20 kV to 120 kV and watch the bremsstrahlung cutoff wavelength \lambda_\text{min} = hc/(eV) shrink as the tube voltage rises. The red curve plots \lambda_\text{min} against tube voltage; the two horizontal dashed lines mark the tungsten Kα (20.9 pm) and Kβ (18.4 pm) characteristic wavelengths — the cutoff only crosses these once V is high enough to eject a K-shell electron in the first place.

Worked examples

Applications — why this physics matters beyond the lab

X-rays get used everywhere for two reasons. First, the wavelength is comparable to the spacing between atoms in solids (0.1–0.3 nm), so X-rays diffract off crystals — this is the foundation of X-ray crystallography, which determined the structure of table salt in 1913, DNA in 1953, and most of the proteins in the protein databank since 1960. Second, the photon energy is high enough to pass through materials that stop visible light (flesh, luggage walls, bearing housings) but not high enough to penetrate thick lead or dense bone, so X-rays give you shadow images of interiors.

Indian science and medicine use X-rays at every scale:

• Medical imaging. AIIMS Delhi, Tata Memorial Hospital Mumbai, and every major teaching hospital runs multiple X-ray tubes daily for chest radiographs, dental films, and bone trauma diagnostics. A chest X-ray at a government hospital in Bhopal uses a 90 kV tube running for 50 ms — about 10^{14} photons through the chest, 10^{13} of which make it to the detector and form the image.
• Radiation therapy. Tata Memorial Centre was founded in 1941 specifically for cancer treatment with radiation. Therapy X-rays run at much higher energies than diagnostic ones — 4 to 25 MV from linear accelerators (linacs), not from Coolidge tubes — but the physics is continuous with the Duane-Hunt picture.
• X-ray crystallography. The Indian Institute of Science (IISc) and the Raman Research Institute in Bangalore run crystallography labs that determine the structures of proteins and small molecules. Typical source: a Cu Kα line (154 pm) diffracted off a rotated single crystal. India's synchrotron Indus-2 at RRCAT Indore provides tunable X-ray beams for the same purpose at higher flux.
• X-ray astronomy. ISRO's AstroSat, launched in 2015 and operated by TIFR, carries the Soft X-ray Telescope (SXT), which images astronomical sources in 0.3 to 8 keV photons — the continuous spectrum of active galactic nuclei, the thermal lines of hot gas in supernova remnants, black-hole accretion disks radiating at millions of kelvin. X-ray astronomy requires going above Earth's atmosphere because atmospheric nitrogen and oxygen absorb these wavelengths entirely (which is also why they are safe for you at ground level).
• Airport security and non-destructive testing. The same tube physics, at slightly lower voltages (around 80 kV), is what produces the coloured X-ray images you see on airport baggage scanners. The colours are not real — they are pseudo-colours that indicate estimated material density and effective atomic number, derived from the way different materials absorb different parts of the continuous spectrum.

Common confusions

• "X-rays and gamma rays are different kinds of radiation." They are the same kind of electromagnetic radiation — both are photons. The distinction is origin: photons produced by electron transitions (either bremsstrahlung or characteristic) are called X-rays; photons produced by transitions inside the nucleus are called gamma rays. Their wavelengths overlap around 10 pm; a 100 keV photon from a nuclear transition is a gamma ray, while a 100 keV photon from an X-ray tube is an X-ray. The photon itself does not know which name to wear.
• "The Duane-Hunt limit depends on the target material." It does not. \lambda_\text{min} = hc/(eV) depends only on the tube voltage, not on what the anode is made of. Switching from tungsten to molybdenum changes the characteristic lines (Kα and Kβ) but does not change the bremsstrahlung cutoff at all. The cutoff is a statement about energy conservation, independent of the details of how the electron gets stopped.
• "Characteristic lines are always present." Only if the tube voltage exceeds the relevant shell binding energy. For tungsten (K-shell binding 70 keV), you need V > 70 kV to see the K lines at all. Below that, only the continuum appears — a tungsten tube at 40 kV produces bremsstrahlung but no Kα. Copper K-edge is 8.98 keV, so copper K lines are easy to see at any reasonable voltage.
• "Bremsstrahlung is a purely quantum effect." The smooth continuum appears in classical electromagnetism too — an accelerated charge radiates, full stop, Maxwell 1865. What is quantum about X-rays is the cutoff at \lambda_\text{min}. Classical electromagnetism allows a decelerating electron to radiate arbitrarily short wavelengths (with vanishing intensity) — there is no fundamental upper frequency for a classical radiator. The cutoff exists because the emitted energy comes in packets of hf; no partial photons.
• "\sqrt f is linear in Z for all X-ray lines." Only approximately, and the Moseley constants a and b are specific to each line. Kα has b \approx 1; Kβ has b \approx 1.8; L-series lines have quite different constants. The underlying physics (screening by inner electrons) is the same, but the effective charge seen by the transitioning electron depends on how many intervening shells are present — which varies from transition to transition.

Where this leads next

• Hydrogen spectrum and spectral series — the Bohr transitions that Moseley's law generalises to heavy atoms.
• Energy levels and quantum numbers — the K, L, M shell structure that organises the characteristic X-ray lines.
• Photoelectric effect — the inverse of characteristic X-ray production: a photon ejects an inner electron and creates the same kind of vacancy.
• Photons and wave-particle duality of light — the quantum-of-light idea that Duane-Hunt tests at the extreme.
• The nucleus — composition and size — where X-rays end and gamma rays begin.