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X-rays are high-frequency electromagnetic radiation, conventionally with wavelengths of roughly $0.01$ to $10\mathrm{nm}$. The precise wavelength convention varies with context. Hot accreting gas and energetic electronic processes can emit X-rays, whose short wavelengths make them useful probes of compact objects and electron shells.

For a photon, $E=h\nu =hc/\lambda$, where $h$ is the Planck constant and $c$ is the speed of light. As a calculated example, $\lambda =0.10\mathrm{nm}$ gives $E\simeq 12.4\mathrm{keV}$, using $hc\simeq 1.23984\mathrm{keV}\mathrm{nm}$. The same wavelength can describe different production mechanisms: bremsstrahlung produces a continuous frequency spectrum, whereas Characteristic X-rays arise from discrete electronic changes in chemical atoms. Measuring transmitted X-rays gives radiography; measuring their frequency spectrum gives spectroscopy.

X-ray diffraction is the angular redistribution of X-ray wave intensity through wave interference between scattered waves. In a crystal structure, repeated lattice translations select discrete outgoing directions by the Laue equations. A photographic plate can therefore turn invisible atomic periodicity into a visible constellation of spots. Their geometry constrains the Bravais lattice and its orientation; their wave intensities also depend on the crystal basis structure factor. Determining atomic positions belongs to X-ray crystallography and requires more information than recognizing a spot pattern.

The Laue X-ray diffraction experiment of 1912 established this phenomenon experimentally. The Laue method uses a fixed crystal structure with a range of incident wavelengths; the Bragg crystal reflection experiment of 1913 instead measured angularly resolved signals with a moving detector. These are different ways of finding the same coherent elastic scattering directions.

X-ray fluorescence is Characteristic X-ray emission following excitation by incident photons. In a simple initially unexcited chemical atom, photoionization can eject an inner Electron if the incident photon supplies its binding energy. An outer Electron can then fill the vacancy, releasing an emitted photon with the difference of shell binding energies. Nonradiative relaxation competes with this route, so one absorbed incident photon need not yield one escaping characteristic photon.

Choose a separate stipulated electron shell model with values $B_K=10.0$, $B_L=1.5$ and $B_M=0.3$, all in $\mathrm{keV}$. Ignore subshell splitting, recoil and changes of binding energy during relaxation. The possible shell differences areThese labels denote simplified shell groups, not resolved subshell electronic transitions. With $hc=1.23984\mathrm{keV}\mathrm{nm}$, their calculated wavelengths are respectively $0.145864$, $0.127819$ and $1.03320\mathrm{nm}$. An incident $9.8\mathrm{keV}$ photon exceeds both K-line energies but cannot directly create the K vacancy in this initially unexcited model. A $12.0\mathrm{keV}$ photon can, leaving approximately $2.0\mathrm{keV}$ of photoelectron kinetic energy. This comparison fixes neither the excitation probability nor the radiative branching probability.

The historical X unit was a practical X-ray wavelength unit, approximately $10^{-13}\mathrm{m}$. Its scale used an assigned calcite spacing; relative wavelength measurements could be more precise than absolute determinations. Variation between crystal specimens led to calibration using reference spectral lines. An X unit value therefore cannot simply be read as an exact multiple of $10^{-13}\mathrm{m}$.^[ref]

An angular-scale check determines how an indicated rotation corresponds to an angle. A reference line transfers a wavelength scale. An absolute determination ties that scale to an independently established length standard. These are different calibration tasks. Repeating measurements can reduce random scatter without correcting the common scale: if every inferred $d$ is too large by a chosen $0.1\%$, all inferred absolute wavelengths are too large by $0.1\%$, while ratios using that same $d$ are unchanged.

In August 1896, Pieter Zeeman observed broadened sodium spectral lines at Leiden, initially without resolved components. His lecture describes a slit and concave diffraction grating, then compares the broadening with the sodium doublet separation: about one thirtieth, roughly $0.2$ ångström. This supplied a spectral reference scale, not an absolute magnetic field calibration. Later cadmium photographs displayed resolved transverse triplets and axial doublets.^[ref]

The fuller report specifies a concave diffraction grating of ten-foot radius with 14,438 lines per inch, an adjustable slit and a micrometer eyepiece with vertical crosswire. Salt in a flame between electromagnet poles supplied the sodium spectral lines. Repeated switching broadened and restored them. To test changes in flame temperature or vapour density, Pieter Zeeman examined photon absorption through a heated porcelain tube with water-cooled glass windows. The arc lamp was displaced from the magnet and its light redirected by a mirror to avoid disturbing the arc. Rotating a narrower tube made the absorption spectral lines equally broad at different heights; the magnetic field still broadened their whole length.^[ref]

For the axial test, pierced electromagnet poles let the light travel along the magnetic field. A quarter-wave plate and polarizer selected one broadened edge; rotating the analyser by $90^{\circ }$ or reversing the magnet current exchanged the visible edge. The crosswire made its displacement detectable. This tested electromagnetic polarization as well as line width. For its rough estimate the report compared each side's widening with the sodium doublet separation, obtaining about one fortieth, and stated an approximate magnetic field intensity in historical units. These passages give no absolute field-calibration procedure; the inferred charge-to-mass ratio was explicitly a first approximation.^[ref]

Hendrik Antoon Lorentz's charged-oscillator theory connected the observed splitting to moving electric charge. In modern language, a line-position measurement needs a wavelength or frequency calibration and an independent magnetic field calibration before it determines a magnetic dipole moment. Resolving the individual components adds the transition information used in the worked example below.

Consider an idealized planet that is a sphere of radius $R$, illuminated by parallel sunlight of irradiance $S$, measured on a plane perpendicular to the rays. Its intercepting projected area is $\pi R^2$, whereas its total surface area is $4\pi R^2$. Therefore the incoming power averaged per unit total surface area iswhere $\alpha$ is the broadband Bond albedo. The factor four is a sphere geometry result, not an atmospheric transmission factor. Local sunlight still varies over the illuminated hemisphere and vanishes on the dark side.

Now stipulate a single spatially uniform temperature $T$, no atmosphere, no internal heating, and emissivity one for outgoing thermal radiation. This is a thermal blackbody model using the Stefan–Boltzmann law; the prescribed solar-band reflection is treated separately, so the body is not a perfect absorber at all wavelengths. Let $C>0$ be the effective heat capacity per unit surface area, in $\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. Conservation of energy gives the ordinary differential equationHere $F$ is net heating per unit surface area, in $\mathrm{W}{\mathrm{m}}^{-2}$, and $\sigma$ is the Stefan-Boltzmann constant. Use $\sigma =5.670374419\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$, rounded from the exact constant reported by NIST. ^[ref] The uniform temperature is a model assumption: generally the surface average of $T^4$ differs from the fourth power of the surface-average temperature.

For fixed Bond albedo $\alpha$, radiative equilibrium requires $F(T_{\ast })=0$, givingWith the illustrative solar constant $S=1360\mathrm{W}{\mathrm{m}}^{-2}$, the global incoming average is $340\mathrm{W}{\mathrm{m}}^{-2}$:

Doubling albedo here lowers the equilibrium temperature by $33.231\mathrm{K}$, rather than halving it: the absorbed fraction changes from $0.7$ to $0.4$, and thermal radiation scales as $T^4$. These stipulated Bond albedos are whole-planet examples, not measurements of bare water and ice. The model predicts neither Earth's modern surface temperature nor its response to actual climate change; it omits the greenhouse effect, spatial heat transfer and seasons.

Each fixed-albedo equilibrium is a stable equilibrium: for $T_{\ast }>0$, the derivative is $F^{\prime }(T_{\ast })=-4\sigma T_{\ast }^3<0$. A small temperature increase produces net cooling, and a small decrease produces net heating. This is the simplest expression of the stabilizing Planck feedback. Allowing albedo to fall with temperature adds a competing term, as in ice-ocean albedo feedback.