Editing Bremsstrahlung (section)

=== X-ray tube ===
[[File:TubeSpectrum-en.svg|thumb|Spectrum of the X-rays emitted by an [[X-ray tube]] with a [[rhodium]] target, operated at 60 [[kilovolt|kV]]. The continuous curve is due to bremsstrahlung, and the spikes are [[energy-dispersive X-ray spectroscopy|characteristic K lines]] for rhodium. The curve goes to zero at 21 [[picometer|pm]] in agreement with the [[Duane–Hunt law]], as described in the text.]]
{{main|X-ray tube}}
In an [[X-ray tube]], electrons are accelerated in a vacuum by an [[electric field]] towards a piece of material called the "target". X-rays are emitted as the electrons hit the target.

Already in the early 20th century physicists found out that X-rays consist of two components, one independent of the target material and another with characteristics of [[fluorescence]].<ref name=":0">{{Cite book |last=Eckert |first=Michael |url=https://books.google.com/books?id=kzMPEAAAQBAJ&pg=PA28 |title=Establishing Quantum Physics in Munich: Emergence of Arnold Sommerfeld’s Quantum School |date=2020-12-15 |publisher=Springer Nature |isbn=978-3-030-62034-9 |language=en}}</ref> Now we say that the output spectrum consists of a continuous spectrum of X-rays with additional sharp peaks at certain energies. The former is due to bremsstrahlung, while the latter are [[characteristic x-ray|characteristic X-rays]] associated with the atoms in the target. For this reason, bremsstrahlung in this context is also called '''continuous X-rays'''.<ref>{{cite book|author=S. J. B. Reed|title=Electron Microprobe Analysis and Scanning Electron Microscopy in Geology| url=https://books.google.com/books?id=9-_v4YgpoVMC&pg=PA12|year=2005|publisher=Cambridge University Press|isbn=978-1-139-44638-9|page=12}}</ref> The German term itself was introduced in 1909 by [[Arnold Sommerfeld]] in order to explain the nature of the first variety of X-rays.<ref name=":0" />

The shape of this continuum spectrum is approximately described by [[Kramers' law]].

The formula for Kramers' law is usually given as the distribution of intensity (photon count) <math>I</math> against the [[wavelength]] <math>\lambda</math> of the emitted radiation:<ref name="laguitton">{{cite journal| doi=10.1002/xrs.1300060409| last=Laguitton| first=Daniel| author2=William Parrish | date=1977| title=Experimental Spectral Distribution versus Kramers' Law for Quantitative X-ray Fluorescence by the Fundamental Parameters Method| journal=X-Ray Spectrometry| volume=6| issue=4| pages=201| bibcode=1977XRS.....6..201L}}</ref>
<math display="block"> I(\lambda) \, d\lambda = K \left( \frac{\lambda}{\lambda_{\min}} - 1 \right)\frac{d\lambda}{\lambda^2} </math>

The constant {{math|''K''}} is proportional to the [[atomic number]] of the target element, and <math>\lambda_{\min}</math> is the minimum wavelength given by the [[Duane–Hunt law]].

The spectrum has a sharp cutoff at {{nowrap|<math>\lambda_{\min}</math>,}} which is due to the limited energy of the incoming electrons. For example, if an electron in the tube is accelerated through 60 [[kilovolt|kV]], then it will acquire a kinetic energy of 60 [[electronvolt|keV]], and when it strikes the target it can create X-rays with energy of at most 60 keV, by [[conservation of energy]]. (This upper limit corresponds to the electron coming to a stop by emitting just one X-ray [[photon]]. Usually the electron emits many photons, and each has an energy less than 60 keV.) A photon with energy of at most 60&nbsp;keV has wavelength of at least {{val|21|ul=pm}}, so the continuous X-ray spectrum has exactly that cutoff, as seen in the graph. More generally the formula for the low-wavelength cutoff, the Duane–Hunt law, is:<ref>{{cite book|author1=Rene Van Grieken| author2=Andrzej Markowicz|title=Handbook of X-Ray Spectrometry|url=https://books.google.com/books?id=i_iDRTp75AsC&pg=PA3| year=2001| publisher=CRC Press| isbn=978-0-203-90870-9|page=3}}</ref>
<math display="block">\lambda_\min = \frac{h c}{e V} \approx \frac{1239.8}{V}\,\mathrm{pm/kV}</math>
where {{math|''h''}} is the [[Planck constant]], {{math|''c''}} is the [[speed of light]], {{mvar|V}} is the [[voltage]] that the electrons are accelerated through, {{math|''e''}} is the [[elementary charge]], and {{math|pm}} is [[picometre]]s.