Bremsstrahlung

remsstrahlung (German pronunciation: ˈbʁɛmsˌʃtʁaːlʊŋ ( listen), from bremsen "to brake" and Strahlung "radiation", i.e. "braking radiation" or "deceleration radiation") is electromagnetic radiation produced by the deceleration of a answerable atom if deflected by addition answerable particle, about an electron by an diminutive nucleus. The affective atom loses active energy, which is adapted into a photon because activity is conserved. The appellation is aswell acclimated to accredit to the action of bearing the radiation. Bremsstrahlung has a connected spectrum, which becomes added acute and accouterment against college frequencies if the activity of the accelerated particles is increased.

Strictly speaking, bremsstrahlung refers to any radiation due to the dispatch of a answerable particle, which includes synchrotron radiation; however, it is frequently acclimated in the added attenuated faculty of radiation from electrons endlessly in matter.

Bremsstrahlung emitted from claret is sometimes referred to as free-free radiation. This refers to the actuality that the radiation in this case is created by answerable particles that are chargeless both afore and afterwards the angle (acceleration) that causes the emission.

Dipole approximation

Suppose that a atom of allegation q adventures an dispatch \vec{a} which is beeline with its dispatch \vec{v} (this is the accordant case for beeline accelerators). Then, the relativistic announcement for the angular administration of the bremsstrahlung (considering alone the ascendant dipole radiation contribution), is

\frac{dP(\theta)}{d\Omega} = \frac{q^2 a^2}{16 \pi^2 \epsilon_0 c^3}\frac{{\sin^2{\theta}}}{(1 - \beta \cos{\theta})^5},

area β = v / c and θ is the bend amid \vec{a} and the point of observation.

Integrating over all angles again gives the absolute ability emitted as 1

P_{a \parallel v} = \frac{q^2 a^2 \gamma^6}{6 \pi \epsilon_0 c^3},

where γ is the Lorentz agency .

The accepted announcement for the absolute broadcast ability is2

P = \frac{q^2 \gamma^4}{6 \pi \epsilon_0 c} \left( \dot{\beta}^2 + \frac{(\vec{\beta} \cdot \dot{\vec{\beta}})^2}{1 - \beta^2}\right)

where \dot{\beta} signifies a time acquired of β. Note, this accepted announcement for absolute broadcast ability simplifies to the aloft announcement for the specific case of dispatch alongside to dispatch (\vec{\beta} \cdot \dot{\vec{\beta}} = \beta \dot{\beta}), by acquainted that \dot{\beta} = a/c and γ = (1 − β2) − 1 / 2. For the case of dispatch erect to the dispatch (\vec{\beta} \cdot \dot{\vec{\beta}} = 0 ) (a case that arises in annular atom accelerators accepted as synchrotrons), the absolute ability broadcast reduces to

P_{a \perp v} = \frac{q^2 a^2 \gamma^4 }{6 \pi \epsilon_0 c^3}.

The absolute ability broadcast in the two attached cases is proportional to γ4 (a \perp v) or γ6 (a \parallel v). Since E = γmc2, we see that the absolute broadcast ability goes as m − 4 or m − 6, which accounts for why electrons lose activity to bremsstrahlung radiation abundant added rapidly than added answerable particles (e.g., muons, protons, alpha particles). This is the acumen a TeV activity electron-positron collider (such as the proposed International Beeline Collider) cannot use a annular adit (requiring connected acceleration), while a proton-proton collider (such as the Large Hadron Collider) can advance a annular tunnel. The electrons lose activity due to bremsstrahlung at a amount (m_p/m_e)^4 \approx 10^{13} times college than protons do.

Thermal bremsstrahlung

In a claret the chargeless electrons consistently aftermath bremsstrahlung in collisions with the ions. A complete assay requires accounting for both bifold Coulomb collisions as able-bodied as aggregate (dielectric) behavior. A abundant analysis is accustomed in,3 some of which is abbreviated in ?, while a simplified one is accustomed in.4 In this area we chase Bekefi's dielectric treatment, with collisions included about via the blow wavenumber km.

Consider a compatible plasma, with thermal electrons (distributed according to the Maxwell–Boltzmann administration with the temperature Te). Following Bekefi, the ability ashen body (power per angular abundance breach per volume, chip over the accomplished 4π sr of solid angle, and in both polarizations) of the bremsstrahlung radiated, is affected to be

{dP_\mathrm{Br} \over d\omega} = {8\sqrt 2 \over 3\sqrt\pi} \left1-{\omega_p^2 \over \omega^2}\right^{1/2} \left Z_i^2 n_i n_e r_e^3 \right \left { \frac{(m_ec^2)^{3/2}}{(k_B T_e)^{1/2}} } \right E_1(y) ,

where ωp is the electron claret frequency, ne,ni is the amount body of electrons and ions, re is the classical ambit of electron, me is its mass, kB is the Boltzmann constant, and c is the acceleration of light. The aboriginal affiliated agency is the basis of refraction of a ablaze beachcomber in a plasma, and shows that discharge is abundantly suppressed for ω < ωp (this is the blow action for a ablaze beachcomber in a plasma; in this case the ablaze beachcomber is evanescent). This blueprint appropriately alone applies for ω > ωp. Note that the added affiliated agency has units of 1/volume and the third agency has units of energy, giving the actual absolute units of energy/volume. This blueprint should be summed over ion breed in a multi-species plasma.

The appropriate action E1 is authentic in the exponential basic article, and the unitless abundance y is

y = {1\over 2}{\omega^2 m_e \over k_m^2 k_B T_e}

km is a best or blow wavenumber, arising due to bifold collisions, and can alter with ion species. Roughly, km = 1 / λB if k_B T_e>Z_i^2 E_h (typical in plasmas that are not too cold), area E_h \approx 27.2 eV is the Hartree energy, and \lambda_B=\hbar/(m_e k_B T_e)^{1/2} is the electron thermal de Broglie wavelength. Otherwise, k_m \propto 1/l_c area lc is the classical Coulomb ambit of abutting approach.

For the accepted case km = 1 / λB, we find

y = {1\over2}\left\frac{\hbar\omega}{k_B T_e}\right^2 .

The blueprint for dPBr / dω is approximate, in that it neglects added discharge occurring for ω hardly aloft ωp.

In the absolute y < < 1, we can almost E1 as E_1(y) \approx -\ln y e^\gamma + O(y) area \gamma\approx 0.577 is the Euler–Mascheroni constant. The leading, logarithmic appellation is frequently used, and resembles the Coulomb logarithm that occurs in added collisional claret calculations. For y > e − γ the log appellation is negative, and the approximation is acutely inadequate. Bekefi gives adapted expressions for the logarthmic appellation that bout abundant binary-collision calculations.

The absolute discharge ability density, chip over all frequencies, is

\begin{align} P_\mathrm{Br} &= \int_{\omega_p}^\infty d\omega {dP_\mathrm{Br}\over d\omega} = {16 \over 3} \left Z_i^2 n_i n_e r_e^3 \right \leftm_e c^3 \right k_m G(y_p) \\ G(y_p) &= {1 \over 2\sqrt{\pi}} \int_{y_p}^\infty dy y^{-1/2} \left1-{y_p\over y}\right^{1/2} E_1(y) \\ y_p &= y(\omega=\omega_p) \end{align}

G(yp = 0) = 1 and decreases with yp; it is consistently positive. For km = 1 / λB, we find

P_\mathrm{Br} = {16 \over 3} \left Z_i^2 n_i n_e r_e^3 \right \left {c \over r_e} (m_e c^2 k_B T_e)^{1/2} \right \alpha G(y_p)

The aboriginal affiliated agency has units of 1/volume, while the added has units of power. Note the actualization of the fine-structure connected α due to the breakthrough attributes of λB. In applied units, a frequently acclimated adaptation of this blueprint for G = 1 is 5

P_\mathrm{Br} \textrm{W/m}^3 = {Z_i^2 n_i n_e \over \left7.69 \times 10^{18} \textrm{m}^{-3}\right^2} T_e\textrm{eV}^{1/2} .

This blueprint is 1.59 times the one accustomed above, with the aberration due to data of bifold collisions. Such ambiguity is about bidding by introducing Gaunt agency gB, e.g. in 6 one finds

\epsilon_\mathrm{ff} = 1.4\times 10^{-27} T^{1/2} n_{e} n_{i} Z^{2} g_B\,,

where aggregate is bidding in the CGS units.

edit Relativistic corrections

Relativistic corrections to the discharge of a 30-keV photon by an electron impacting on a proton.

For actual top temperatures there are relativistic corrections to this formula, that is, added agreement of the adjustment of k_B T_e/m_e c^2\,.1

edit Bremsstrahlung cooling

If the claret is optically thin, the bremsstrahlung radiation leaves the plasma, accustomed allotment of the centralized claret energy. This aftereffect is accepted as the bremsstrahlung cooling. It is a blazon of radiative cooling. The activity agitated abroad by bremsstrahlung is alleged bremsstrahlung losses and represent, respectively, a blazon of radiative losses. One about uses the appellation bremsstrahlung losses in the ambience if the claret cooling is undesired, as e.g. in admixture plasmas.

Sources of bremsstrahlung

X-ray tube

Spectrum of the X-rays emitted by an X-ray tube with a rhodium target, operated at 60 kV. The connected ambit is due to bremsstrahlung, and the spikes are appropriate K ambit for rhodium. The ambit goes to aught at 21 message in acceding with the Duane–Hunt law, as declared in the text.

Main article: X-ray

In an X-ray tube, electrons are accelerated in a exhaustion by an electric acreage and attempt into a section of metal alleged the "target". X-rays are emitted as the electrons apathetic down (decelerate) in the metal. The achievement spectrum consists of a connected spectrum of X-rays, with added aciculate peaks at assertive energies (see blueprint on right). The connected spectrum is due to bremsstrahlung, while the aciculate peaks are appropriate X-rays associated with the atoms in the target. For this reason, bremsstrahlung in this ambience is aswell alleged connected X-rays.7

The spectrum has a aciculate blow at low wavelength, which is due to the bound activity of the admission electrons. For example, if an electron in the tube is accelerated through 60 kV, again it will access a active activity of 60 keV, and if it strikes the ambition it can actualize X-rays with activity of at a lot of 60 keV, by attention of energy. (This high absolute corresponds to the electron advancing to a stop by abounding just one X-ray photon. Usually the electron emits abounding photons, and anniversary has an activity beneath than 60 keV.) A photon with activity of at a lot of 60 keV has amicableness of at atomic 21 pm, so the connected X-ray spectrum has absolutely that cutoff, as apparent in the graph. More about the blueprint for the low-wavelength blow is:8

\lambda_{\text{min}} = \frac{h c}{e V} \approx \frac{1239.8 \text{ pm}}{V\text{ in kV}} \,

where h is Planck constant, c is the acceleration of light, V is the voltage that the electrons are accelerated through, e is the elementary charge, and message is picometres. This is alleged the Duane–Hunt law.

edit Beta decay

Main article: Beta decay

Beta particle-emitting substances sometimes display a anemic radiation with connected spectrum that is due to bremsstrahlung. In this context, bremsstrahlung is a blazon of "secondary radiation", in that it is produced as a aftereffect of endlessly (or slowing) the primary radiation (beta particles). In electron and positron discharge the photon's activity comes from the electron/nucleon pair, with the spectrum of the bremsstrahlung abbreviating continuously with accretion activity of the beta particle. In electron abduction the activity comes at the amount of the neutrino, and the spectrum is greatest at about one third of the accustomed neutrino energy, extensive aught at aught activity and at accustomed neutrino energy.

edit Inner and alien bremsstrahlung

The "inner" bremsstrahlung arises from the conception of the electron and its accident of activity (due to the able electric acreage in the arena of the basis ability decay) as it leaves the nucleus. This is to be assorted with the "outer" bremsstrahlung due to the approximation on the basis of electrons advancing from the alfresco (i.e., emitted by addition nucleus).9

edit Radiation safety

In some cases, e.g. 32P, the bremsstrahlung produced by careful the beta radiation with the commonly acclimated close abstracts (e.g. lead) is itself dangerous; in such cases, careful have to be able with low body materials, e.g. Plexiglass (lucite), plastic, wood, or water;10 because the amount of deceleration of the electron is slower, the radiation accustomed off has a best amicableness and is accordingly beneath penetrating.

In astrophysics

The ascendant beaming basic in a array of galaxies is the 107 to 108 kelvin intracluster medium. The discharge from the intracluster average is characterized by thermal bremsstrahlung. This radiation is in the activity ambit of X-rays and can be calmly empiric with space-based telescopes such as Chandra X-ray Observatory, XMM-Newton, ROSAT, ASCA, EXOSAT, Suzaku, RHESSI and approaching missions like IXO 2 and Astro-H 3.

Bremsstrahlung is aswell the ascendant discharge apparatus for H II regions at radio wavelengths.