Bremsstrahlung

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Bremsstrahlung produced by a high-energy electron deflected in the electric field of an atomic nucleus.

Bremsstrahlung /ˈbrɛmʃtrɑːləŋ/, from bremsen "to brake" and Strahlung "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation (i.e., photons), thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.

Broadly speaking, bremsstrahlung or braking radiation is any radiation produced due to the deceleration (negative acceleration) of a charged particle, which includes synchrotron radiation (i.e., photon emission by a relativistic particle), cyclotron radiation (i.e. photon emission by a non-relativistic particle), and the emission of electrons and positrons during beta decay. However, the term is frequently used in the more narrow sense of radiation from electrons (from whatever source) slowing in matter.

Bremsstrahlung emitted from plasma is sometimes referred to as free–free radiation. This refers to the fact that the radiation in this case is created by electrons that are free (i.e., not in an atomic or molecular bound state) before, and remain free after, the emission of a photon. In the same parlance, bound–bound radiation refers to discrete spectral lines (an electron "jumps" between two bound states), while free–bound radiation refers to the radiative combination process, in which a free electron recombines with an ion.

Classical description

Main article: Larmor formula

Field lines and modulus of the electric field generated by a (negative) charge first moving at a constant speed and then stopping quickly to show the generated Bremsstrahlung radiation.

If quantum effects are negligible, an accelerating charged particle radiates power as described by the Larmor formula and its relativistic generalization.

Total radiated power

The total radiated power is

${\displaystyle P={\frac {q^{2}\gamma ^{4}}{6\pi \varepsilon _{0}c}}\left({\dot {\beta }}^{2}+{\frac {\left({\boldsymbol {\beta }}\cdot {\dot {\boldsymbol {\beta }}}\right)^{2}}{1-\beta ^{2}}}\right),}$

where ${\textstyle {\boldsymbol {\beta }}={\frac {\mathbf {v} }{c}}}$ (the velocity of the particle divided by the speed of light), ${\textstyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}$ is the Lorentz factor, ${\displaystyle {\dot {\boldsymbol {\beta }}}}$ signifies a time derivative of ${\displaystyle {\boldsymbol {\beta }}}$, and q is the charge of the particle. In the case where velocity is parallel to acceleration (i.e., linear motion), the expression reduces to

${\displaystyle P_{a\parallel v}={\frac {q^{2}a^{2}\gamma ^{6}}{6\pi \varepsilon _{0}c^{3}}},}$

where ${\displaystyle a\equiv {\dot {v}}={\dot {\beta }}c}$ is the acceleration. For the case of acceleration perpendicular to the velocity (${\displaystyle {\boldsymbol {\beta }}\cdot {\dot {\boldsymbol {\beta }}}=0}$), for example in synchrotrons, the total power is

${\displaystyle P_{a\perp v}={\frac {q^{2}a^{2}\gamma ^{4}}{6\pi \varepsilon _{0}c^{3}}}.}$

Power radiated in the two limiting cases is proportional to ${\displaystyle \gamma ^{4}}$ ${\displaystyle \left(a\perp v\right)}$ or ${\displaystyle \gamma ^{6}}$ ${\displaystyle \left(a\parallel v\right)}$. Since ${\displaystyle E=\gamma mc^{2}}$, we see that for particles with the same energy ${\displaystyle E}$ the total radiated power goes as ${\displaystyle m^{-4}}$ or ${\displaystyle m^{-6}}$, which accounts for why electrons lose energy to bremsstrahlung radiation much more rapidly than heavier charged particles (e.g., muons, protons, alpha particles). This is the reason a TeV energy electron-positron collider (such as the proposed International Linear Collider) cannot use a circular tunnel (requiring constant acceleration), while a proton-proton collider (such as the Large Hadron Collider) can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate ${\displaystyle (m_{p}/m_{e})^{4}\approx 10^{13}}$ times higher than protons do.

Angular distribution

The most general formula for radiated power as a function of angle is:

${\displaystyle {\frac {dP}{d\Omega }}={\frac {q^{2}}{16\pi ^{2}\varepsilon _{0}c}}{\frac {\left|{\hat {\mathbf {n} }}\times \left(\left({\hat {\mathbf {n} }}-{\boldsymbol {\beta }}\right)\times {\dot {\boldsymbol {\beta }}}\right)\right|^{2}}{\left(1-{\hat {\mathbf {n} }}\cdot {\boldsymbol {\beta }}\right)^{5}}}}$,

where ${\displaystyle {\hat {\mathbf {n} }}}$ is a unit vector pointing from the particle towards the observer, and ${\displaystyle d\Omega }$ is an infinitesimal bit of solid angle.

In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to

${\displaystyle {\frac {dP_{a\parallel v}}{d\Omega }}={\frac {q^{2}a^{2}}{16\pi ^{2}\varepsilon _{0}c^{3}}}{\frac {\sin ^{2}\theta }{(1-\beta \cos \theta )^{5}}}}$,

where ${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbf {a} }$ and the direction of observation.

Simplified quantum description

This section gives a quantum-mechanical analog of the prior section, but with some simplifications. We give a non-relativistic treatment of the special case of an electron of mass ${\displaystyle m_{e}}$, charge ${\displaystyle -e}$, and initial speed ${\displaystyle v}$ decelerating in the Coulomb field of a gas of heavy ions of charge ${\displaystyle Ze}$ and number density ${\displaystyle n_{i}}$. The emitted radiation is a photon of frequency ${\displaystyle \nu =c/\lambda }$ and energy ${\displaystyle h\nu }$. We wish to find the emissivity ${\displaystyle j(v,\nu )}$ which is the power emitted per (solid angle in photon velocity space * photon frequency), summed over both transverse photon polarizations. We follow the common astrophysical practice of writing this result in terms of an approximate classical result times the free-free emission Gaunt factor g_ff which incorporates quantum and other corrections:

$${\displaystyle j(v,\nu )={8\pi \over 3{\sqrt {3}}}\left({e^{2} \over 4\pi \epsilon _{0}}\right)^{3}{Z^{2}n_{i} \over c^{3}m_{e}^{2}v}g_{\rm {ff}}(v,\nu )}$$

.

A general, quantum-mechanical formula for ${\displaystyle g_{\rm {ff}}}$ exists but is very complicated, and usually is found by numerical calculations. We present some approximate results with the following additional assumptions:

• Vacuum interaction: we neglect any effects of the background medium, such as plasma screening effects. This is reasonable for photon frequency much greater than the plasma frequency ${\displaystyle \nu _{\rm {pe}}\propto n_{\rm {e}}^{1/2}}$with ${\displaystyle n_{e}}$ the plasma electron density. Note that light waves are evanescent for ${\displaystyle \nu <\nu _{\rm {pe}}}$ and a significantly different approach would be needed.
• Soft photons: ${\displaystyle h\nu \ll m_{e}v^{2}/2}$, that is, the photon energy is much less than the initial electron kinetic energy.

With these assumptions, two unitless parameters characterize the process: ${\displaystyle \eta _{Z}\equiv Ze^{2}/\hbar v}$, which measures the strength of the electron-ion Coulomb interaction, and ${\displaystyle \eta _{\nu }\equiv h\nu /2m_{e}v^{2}}$, which measures the photon "softness" and we assume is always small (the choice of the factor 2 is for later convenience). In the limit ${\displaystyle \eta _{Z}\ll 1}$, the quantum-mechanical Born approximation gives:

$${\displaystyle g_{\rm {ff,Born}}={{\sqrt {3}} \over \pi }\ln {1 \over \eta _{\nu }}}$$

.

In the opposite limit ${\displaystyle \eta _{Z}\gg 1}$, the full quantum-mechanical result reduces to the purely classical result

$${\displaystyle g_{\rm {ff,class}}={{\sqrt {3}} \over \pi }\left[\ln \left({1 \over \eta _{Z}\eta _{\nu }}\right)-\gamma \right]}$$

,

where ${\displaystyle \gamma \approx 0.577}$ is the Euler–Mascheroni constant. Note that ${\displaystyle 1/\eta _{Z}\eta _{\nu }=m_{e}v^{3}/\pi Ze^{2}\nu }$ which is a purely classical expression without Planck's constant ${\displaystyle h}$.

A semi-classical, heuristic way to understand the Gaunt factor is to write it as ${\displaystyle g_{\rm {ff}}\approx \ln(b_{\rm {max}}/b_{\rm {min}})}$ where ${\displaystyle b_{\max }}$ and ${\displaystyle b_{\rm {min}}}$ are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, ${\displaystyle b_{\rm {max}}=v/\nu }$: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. ${\displaystyle b_{\rm {min}}}$ is the larger of the quantum-mechanical deBroglie wavelength ${\displaystyle \approx h/m_{e}v}$ and the classical distance of closest approach ${\displaystyle \approx e^{2}/4\pi \epsilon _{0}m_{e}v^{2}}$ where the electron-ion Coulomb potential energy is comparable to the electron's initial kinetic energy.

The above results generally apply as long as the argument of the logarithm is large, and break down when it is less than unity. Namely, the Gaunt factor becomes negative in this case, which is unphysical. A rough approximation to the full calculations, with the appropriate Born and classical limits, is

$${\displaystyle g_{\rm {ff}}\approx \max \left[1,{{\sqrt {3}} \over \pi }\ln \left[{1 \over \eta _{\nu }\max(1,e^{\gamma }\eta _{Z})}\right]\right]}$$

.

Thermal bremsstrahlung: emission and absorption

The bremsstrahlung power spectrum rapidly decreases for large ${\displaystyle \omega }$, and is also suppressed near ${\displaystyle \omega =\omega _{\rm {p}}}$. This plot is for the quantum case ${\displaystyle T_{e}>Z^{2}E_{\rm {h}}}$, and ${\displaystyle \hbar \omega _{\rm {p}}/T_{e}=0.1}$.

This section discusses bremsstrahlung emission and the inverse absorption process (called inverse bremsstrahlung) in a macroscopic medium. We start with the equation of radiative transfer, which applies to general processes and not just bremsstrahlung:

$${\displaystyle {\frac {1}{c}}\partial _{t}I_{\nu }+{\hat {\mathbf {n} }}\cdot \nabla I_{\nu }=j_{\nu }-k_{\nu }I_{\nu }}$$

${\displaystyle I_{\nu }(t,\mathbf {x} )}$ is the radiation spectral intensity, or power per (area * solid angle in photon velocity space * photon frequency) summed over both polarizations. ${\displaystyle j_{\nu }}$ is the emissivity, analogous to ${\displaystyle j(v,\nu )}$defined above, and ${\displaystyle k_{\nu }}$ is the absorptivity. ${\displaystyle j_{\nu }}$ and ${\displaystyle k_{\nu }}$ are properties of the matter, not the radiation, and account for all the particles in the medium - not just a pair of one electron and one ion as in the prior section. If ${\displaystyle I_{\nu }}$ is uniform in space and time, then the left-hand side of the transfer equation is zero, and we find

$${\displaystyle I_{\nu }={j_{\nu } \over k_{\nu }}}$$

If the matter and radiation are also in thermal equilibrium at some temperature, then ${\displaystyle I_{\nu }}$must be the blackbody spectrum:

$${\displaystyle B_{\nu }(\nu ,T_{e})={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /k_{\rm {B}}T_{e}}-1}}}$$

Since ${\displaystyle j_{\nu }}$ and ${\displaystyle k_{\nu }}$ are independent of ${\displaystyle I_{\nu }}$, this means that ${\displaystyle j_{\nu }/k_{\nu }}$ must be the blackbody spectrum whenever the matter is in equilibrium at some temperature – regardless of the state of the radiation. This allows us to immediately know both ${\displaystyle j_{\nu }}$ and ${\displaystyle k_{\nu }}$ once one is known – for matter in equilibrium.

In plasma

NOTE: this section currently gives formulas that apply in the Rayleigh-Jeans limit ${\displaystyle \hbar \omega \ll k_{\rm {B}}T_{e}}$, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like ${\displaystyle \exp(-\hbar \omega /k_{\rm {B}}T_{e})}$ does not appear. The appearance of ${\displaystyle \hbar \omega /k_{\rm {B}}T_{e}}$ in ${\displaystyle y}$ below is due to the quantum-mechanical treatment of collisions.

In a plasma, the free electrons continually collide with the ions, producing bremsstrahlung. A complete analysis requires accounting for both binary Coulomb collisions as well as collective (dielectric) behavior. A detailed treatment is given by Bekefi, while a simplified one is given by Ichimaru. In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber, ${\displaystyle k_{\rm {max}}}$.

Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature ${\displaystyle T_{e}}$. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole ${\displaystyle 4\pi }$ sr of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be

${\displaystyle {dP_{\mathrm {Br} } \over d\omega }={8{\sqrt {2}} \over 3{\sqrt {\pi }}}\left[{e^{2} \over 4\pi \varepsilon _{0}}\right]^{3}{1 \over (m_{e}c^{2})^{3/2}}\left[1-{\omega _{\rm {p}}^{2} \over \omega ^{2}}\right]^{1/2}{Z_{i}^{2}n_{i}n_{e} \over (k_{\rm {B}}T_{e})^{1/2}}E_{1}(y),}$

where ${\displaystyle \omega _{p}\equiv (n_{e}e^{2}/\varepsilon _{0}m_{e})^{1/2}}$ is the electron plasma frequency, ${\displaystyle \omega }$ is the photon frequency, ${\displaystyle n_{e},n_{i}}$ is the number density of electrons and ions, and other symbols are physical constants. The second bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed for ${\displaystyle \omega <\omega _{\rm {p}}}$ (this is the cutoff condition for a light wave in a plasma; in this case the light wave is evanescent). This formula thus only applies for ${\displaystyle \omega >\omega _{\rm {p}}}$. This formula should be summed over ion species in a multi-species plasma.

The special function ${\displaystyle E_{1}}$ is defined in the exponential integral article, and the unitless quantity ${\displaystyle y}$ is

${\displaystyle y={1 \over 2}{\omega ^{2}m_{e} \over k_{\rm {max}}^{2}k_{\rm {B}}T_{e}}}$.

${\displaystyle k_{\rm {max}}}$ is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, ${\displaystyle k_{\rm {max}}=1/\lambda _{\rm {B}}}$ when ${\displaystyle k_{\rm {B}}T_{\rm {e}}>Z_{i}^{2}E_{\rm {h}}}$ (typical in plasmas that are not too cold), where ${\displaystyle E_{\rm {h}}\approx 27.2}$ eV is the Hartree energy, and ${\displaystyle \lambda _{\rm {B}}=\hbar /(m_{\rm {e}}k_{\rm {B}}T_{\rm {e}})^{1/2}}$ is the electron thermal de Broglie wavelength. Otherwise, ${\displaystyle k_{\rm {max}}\propto 1/l_{\rm {C}}}$ where ${\displaystyle l_{\rm {C}}}$ is the classical Coulomb distance of closest approach.

For the usual case ${\displaystyle k_{m}=1/\lambda _{B}}$, we find

${\displaystyle y={1 \over 2}\left[{\frac {\hbar \omega }{k_{\rm {B}}T_{e}}}\right]^{2}.}$

The formula for ${\displaystyle dP_{\mathrm {Br} }/d\omega }$ is approximate, in that it neglects enhanced emission occurring for ${\displaystyle \omega }$ slightly above ${\displaystyle \omega _{\rm {p}}}$.

In the limit ${\displaystyle y\ll 1}$, we can approximate ${\displaystyle E_{1}}$ as ${\displaystyle E_{1}(y)\approx -\ln[ye^{\gamma }]+O(y)}$ where ${\displaystyle \gamma \approx 0.577}$ is the Euler–Mascheroni constant. The leading, logarithmic term is frequently used, and resembles the Coulomb logarithm that occurs in other collisional plasma calculations. For ${\displaystyle y>e^{-\gamma }}$ the log term is negative, and the approximation is clearly inadequate. Bekefi gives corrected expressions for the logarithmic term that match detailed binary-collision calculations.

The total emission power density, integrated over all frequencies, is

${\displaystyle {\begin{aligned}P_{\mathrm {Br} }&=\int _{\omega _{\rm {p}}}^{\infty }d\omega {dP_{\mathrm {Br} } \over d\omega }={16 \over 3}\left[{e^{2} \over 4\pi \varepsilon _{0}}\right]^{3}{1 \over m_{e}^{2}c^{3}}Z_{i}^{2}n_{i}n_{e}k_{\rm {max}}G(y_{\rm {p}})\\G(y_{p})&={1 \over 2{\sqrt {\pi }}}\int _{y_{\rm {p}}}^{\infty }dy\,y^{-{\frac {1}{2}}}\left[1-{y_{\rm {p}} \over y}\right]^{\frac {1}{2}}E_{1}(y)\\y_{\rm {p}}&=y(\omega =\omega _{\rm {p}})\end{aligned}}}$

${\displaystyle G(y_{\rm {p}}=0)=1}$ and decreases with ${\displaystyle y_{\rm {p}}}$; it is always positive. For ${\displaystyle k_{\rm {max}}=1/\lambda _{\rm {B}}}$, we find

${\displaystyle P_{\mathrm {Br} }={16 \over 3}{\left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{3} \over (m_{e}c^{2})^{\frac {3}{2}}\hbar }Z_{i}^{2}n_{i}n_{e}(k_{\rm {B}}T_{e})^{\frac {1}{2}}G(y_{\rm {p}})}$

Note the appearance of ${\displaystyle \hbar }$ due to the quantum nature of ${\displaystyle \lambda _{\rm {B}}}$. In practical units, a commonly used version of this formula for ${\displaystyle G=1}$ is 

${\displaystyle P_{\mathrm {Br} }[{\textrm {W}}/{\textrm {m}}^{3}]={Z_{i}^{2}n_{i}n_{e} \over \left[7.69\times 10^{18}{\textrm {m}}^{-3}\right]^{2}}T_{e}[{\textrm {eV}}]^{\frac {1}{2}}.}$

This formula is 1.59 times the one given above, with the difference due to details of binary collisions. Such ambiguity is often expressed by introducing Gaunt factor ${\displaystyle g_{\rm {B}}}$, e.g. in one finds

${\displaystyle \varepsilon _{\mathrm {ff} }=1.4\times 10^{-27}T^{\frac {1}{2}}n_{e}n_{i}Z^{2}g_{\rm {B}},\,}$

where everything is expressed in the CGS units.

Relativistic corrections

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

For very high temperatures there are relativistic corrections to this formula, that is, additional terms of the order of ${\displaystyle k_{\rm {B}}T_{e}/m_{e}c^{2}\,.}$

Bremsstrahlung cooling

If the plasma is optically thin, the bremsstrahlung radiation leaves the plasma, carrying part of the internal plasma energy. This effect is known as the bremsstrahlung cooling. It is a type of radiative cooling. The energy carried away by bremsstrahlung is called bremsstrahlung losses and represents a type of radiative losses. One generally uses the term bremsstrahlung losses in the context when the plasma cooling is undesired, as e.g. in fusion plasmas.

Polarizational bremsstrahlung

Polarizational bremsstrahlung (sometimes referred to as "atomic bremsstrahlung") is the radiation emitted by the target's atomic electrons as the target atom is polarized by the Coulomb field of the incident charged particle. Polarizational bremsstrahlung contributions to the total bremsstrahlung spectrum have been observed in experiments involving relatively massive incident particles, resonance processes, and free atoms. However, there is still some debate as to whether or not there are significant polarizational bremsstrahlung contributions in experiments involving fast electrons incident on solid targets.

It is worth noting that the term "polarizational" is not meant to imply that the emitted bremsstrahlung is polarized. Also, the angular distribution of polarizational bremsstrahlung is theoretically quite different than ordinary bremsstrahlung.

Sources

X-ray tube

Spectrum of the X-rays emitted by an X-ray tube with a rhodium target, operated at 60 kV. The continuous curve is due to bremsstrahlung, and the spikes are characteristic K lines for rhodium. The curve goes to zero at 21 pm in agreement with the Duane–Hunt law, as described in the text.

 

Main article: X-ray tube

In an X-ray tube, electrons are accelerated in a vacuum by an electric field towards a piece of metal called the "target". X-rays are emitted as the electrons slow down (decelerate) in the metal. The output spectrum consists of a continuous spectrum of X-rays, with additional sharp peaks at certain energies. The continuous spectrum is due to bremsstrahlung, while the sharp peaks are characteristic X-rays associated with the atoms in the target. For this reason, bremsstrahlung in this context is also called continuous X-rays.

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) ${\displaystyle I}$ against the wavelength ${\displaystyle \lambda }$ of the emitted radiation:

${\displaystyle I(\lambda )\,d\lambda =K\left({\frac {\lambda }{\lambda _{\min }}}-1\right){\frac {1}{\lambda ^{2}}}\,d\lambda }$

The constant K is proportional to the atomic number of the target element, and ${\displaystyle \lambda _{\min }}$ is the minimum wavelength given by the Duane–Hunt law.

The spectrum has a sharp cutoff at ${\displaystyle \lambda _{\min }}$, which is due to the limited energy of the incoming electrons. For example, if an electron in the tube is accelerated through 60 kV, then it will acquire a kinetic energy of 60 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 keV has wavelength of at least 21 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:

${\displaystyle \lambda _{\min }={\frac {hc}{eV}}\approx {\frac {1239.8}{V}}{\text{ pm/kV}}}$

where h is Planck's constant, c is the speed of light, V is the voltage that the electrons are accelerated through, e is the elementary charge, and pm is picometres.

Beta decay

Main article: Beta decay

Beta particle-emitting substances sometimes exhibit a weak radiation with continuous spectrum that is due to bremsstrahlung (see the "outer bremsstrahlung" below). In this context, bremsstrahlung is a type of "secondary radiation", in that it is produced as a result of stopping (or slowing) the primary radiation (beta particles). It is very similar to X-rays produced by bombarding metal targets with electrons in X-ray generators (as above) except that it is produced by high-speed electrons from beta radiation.

Inner and outer bremsstrahlung

The "inner" bremsstrahlung (also known as "internal bremsstrahlung") arises from the creation of the electron and its loss of energy (due to the strong electric field in the region of the nucleus undergoing decay) as it leaves the nucleus. Such radiation is a feature of beta decay in nuclei, but it is occasionally (less commonly) seen in the beta decay of free neutrons to protons, where it is created as the beta electron leaves the proton.

In electron and positron emission by beta decay the photon's energy comes from the electron-nucleon pair, with the spectrum of the bremsstrahlung decreasing continuously with increasing energy of the beta particle. In electron capture, the energy comes at the expense of the neutrino, and the spectrum is greatest at about one third of the normal neutrino energy, decreasing to zero electromagnetic energy at normal neutrino energy. Note that in the case of electron capture, bremsstrahlung is emitted even though no charged particle is emitted. Instead, the bremsstrahlung radiation may be thought of as being created as the captured electron is accelerated toward being absorbed. Such radiation may be at frequencies that are the same as soft gamma radiation, but it exhibits none of the sharp spectral lines of gamma decay, and thus is not technically gamma radiation.

The internal process is to be contrasted with the "outer" bremsstrahlung due to the impingement on the nucleus of electrons coming from the outside (i.e., emitted by another nucleus), as discussed above.

Radiation safety

In some cases, e.g. ³²
P, the bremsstrahlung produced by shielding the beta radiation with the normally used dense materials (e.g. lead) is itself dangerous; in such cases, shielding must be accomplished with low density materials, e.g. Plexiglas (Lucite), plastic, wood, or water; as the atomic number is lower for these materials, the intensity of bremsstrahlung is significantly reduced, but a larger thickness of shielding is required to stop the electrons (beta radiation).

In astrophysics

The dominant luminous component in a cluster of galaxies is the 10⁷ to 10⁸ kelvin intracluster medium. The emission from the intracluster medium is characterized by thermal bremsstrahlung. This radiation is in the energy range of X-rays and can be easily observed with space-based telescopes such as Chandra X-ray Observatory, XMM-Newton, ROSAT, ASCA, EXOSAT, Suzaku, RHESSI and future missions like IXO and Astro-H.

Bremsstrahlung is also the dominant emission mechanism for H II regions at radio wavelengths.

In electric discharges

In electric discharges, for example as laboratory discharges between two electrodes or as lightning discharges between cloud and ground or within clouds, electrons produce Bremsstrahlung photons while scattering off air molecules. These photons become manifest in terrestrial gamma-ray flashes and are the source for beams of electrons, positrons, neutrons and protons. The appearance of Bremsstrahlung photons also influences the propagation and morphology of discharges in nitrogen-oxygen mixtures with low percentages of oxygen.

Quantum mechanical description

The complete quantum mechanical description was first performed by Bethe and Heitler. They assumed plane waves for electrons which scatter at the nucleus of an atom, and derived a cross section which relates the complete geometry of that process to the frequency of the emitted photon. The quadruply differential cross section which shows a quantum mechanical symmetry to pair production, is:

${\displaystyle {\begin{aligned}d^{4}\sigma ={}&{\frac {Z^{2}\alpha _{\text{fine}}^{3}\hbar ^{2}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {d\omega }{\omega }}{\frac {d\Omega _{i}\,d\Omega _{f}\,d\Phi }{\left|\mathbf {q} \right|^{4}}}\\&{}\times \left[{\frac {\mathbf {p} _{f}^{2}\sin ^{2}\Theta _{f}}{\left(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f}\right)^{2}}}\left(4E_{i}^{2}-c^{2}\mathbf {q} ^{2}\right)+{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\Theta _{i}}{\left(E_{i}-c\left|\mathbf {p} _{i}\right|\cos \Theta _{i}\right)^{2}}}\left(4E_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right.\\&{}\qquad +2\hbar ^{2}\omega ^{2}{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\Theta _{i}+\mathbf {p} _{f}^{2}\sin ^{2}\Theta _{f}}{(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f})\left(E_{i}-c\left|\mathbf {p} _{i}\right|\cos \Theta _{i}\right)}}\\&{}\qquad -2\left.{\frac {\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\sin \Theta _{i}\sin \Theta _{f}\cos \Phi }{\left(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f}\right)\left(E_{i}-c\left|\mathbf {p} _{i}\right|c1\cos \Theta _{i}\right)}}\left(2E_{i}^{2}+2E_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right].\end{aligned}}}$

There ${\displaystyle Z}$ is the atomic number, ${\displaystyle \alpha _{\text{fine}}\approx 1/137}$ the fine structure constant, ${\displaystyle \hbar }$ the reduced Planck's constant and ${\displaystyle c}$ the speed of light. The kinetic energy ${\displaystyle E_{{\text{kin}},i/f}}$ of the electron in the initial and final state is connected to its total energy ${\displaystyle E_{i,f}}$ or its momenta ${\displaystyle \mathbf {p} _{i,f}}$ via

${\displaystyle E_{i,f}=E_{{\text{kin}},i/f}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+\mathbf {p} _{i,f}^{2}c^{2}}},}$

where ${\displaystyle m_{e}}$ is the mass of an electron. Conservation of energy gives

${\displaystyle E_{f}=E_{i}-\hbar \omega ,}$

where ${\displaystyle \hbar \omega }$ is the photon energy. The directions of the emitted photon and the scattered electron are given by

${\displaystyle {\begin{aligned}\Theta _{i}&=\sphericalangle (\mathbf {p} _{i},\mathbf {k} ),\\\Theta _{f}&=\sphericalangle (\mathbf {p} _{f},\mathbf {k} ),\\\Phi &={\text{Angle between the planes }}(\mathbf {p} _{i},\mathbf {k} ){\text{ and }}(\mathbf {p} _{f},\mathbf {k} ),\end{aligned}}}$

where ${\displaystyle \mathbf {k} }$ is the momentum of the photon.

The differentials are given as

${\displaystyle {\begin{aligned}d\Omega _{i}&=\sin \Theta _{i}\ d\Theta _{i},\\d\Omega _{f}&=\sin \Theta _{f}\ d\Theta _{f}.\end{aligned}}}$

The absolute value of the virtual photon between the nucleus and electron is

${\displaystyle {\begin{aligned}-\mathbf {q} ^{2}={}&-\left|\mathbf {p} _{i}\right|^{2}-\left|\mathbf {p} _{f}\right|^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2\left|\mathbf {p} _{i}\right|{\frac {\hbar }{c}}\omega \cos \Theta _{i}-2\left|\mathbf {p} _{f}\right|{\frac {\hbar }{c}}\omega \cos \Theta _{f}\\&{}+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\left(\cos \Theta _{f}\cos \Theta _{i}+\sin \Theta _{f}\sin \Theta _{i}\cos \Phi \right).\end{aligned}}}$

The range of validity is given by the Born approximation

${\displaystyle v\gg {\frac {Zc}{137}}}$

where this relation has to be fulfilled for the velocity ${\displaystyle v}$ of the electron in the initial and final state.

For practical applications (e.g. in Monte Carlo codes) it can be interesting to focus on the relation between the frequency ${\displaystyle \omega }$ of the emitted photon and the angle between this photon and the incident electron. Köhn and Ebert integrated the quadruply differential cross section by Bethe and Heitler over ${\displaystyle \Phi }$ and ${\displaystyle \Theta _{f}}$ and obtained:

${\displaystyle {\frac {d^{2}\sigma (E_{i},\omega ,\Theta _{i})}{d\omega \,d\Omega _{i}}}=\sum \limits _{j=1}^{6}I_{j}}$

with

${\displaystyle {\begin{aligned}I_{1}={}&{\frac {2\pi A}{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\ln \left({\frac {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\left(\Delta _{1}+\Delta _{2}\right)+\Delta _{1}\Delta _{2}}{-\Delta _{2}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\left(\Delta _{1}-\Delta _{2}\right)+\Delta _{1}\Delta _{2}}}\right)\\&{}\times \left[1+{\frac {c\Delta _{2}}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}-{\frac {p_{i}^{2}c^{2}\sin ^{2}\Theta _{i}}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{f}\Delta _{2}}{c\left(E_{i}-cp_{i}\cos \Theta _{i}\right)\left(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\right],\\I_{2}={}&-{\frac {2\pi Ac}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}\ln \left({\frac {E_{f}+p_{f}c}{E_{f}-p_{f}c}}\right),\\I_{3}={}&{\frac {2\pi A}{\sqrt {\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{4}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\times \ln \left[\left(\left[E_{f}+p_{f}c\right]\right.\right.\\&\left.\left[4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left(E_{f}-p_{f}c\right)+\left(\Delta _{1}+\Delta _{2}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]-{\sqrt {\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right)\right]\right)\\&\left[\left(E_{f}-p_{f}c\right)\left(4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[-E_{f}-p_{f}c\right]\right.\right.\\&{}+\left.\left.\left(\Delta _{1}-\Delta _{2}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]-{\sqrt {\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right]\right)\right]^{-1}\\&{}\times \left[-{\frac {\left(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(E_{f}^{3}+E_{f}p_{f}^{2}c^{2}\right)+p_{f}c\left(2\left[\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right]E_{f}p_{f}c+\Delta _{1}\Delta _{2}\left[3E_{f}^{2}+p_{f}^{2}c^{2}\right]\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right.\\&{}-{\frac {c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}-{\frac {4E_{i}^{2}p_{f}^{2}\left(2\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}-4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(\Delta _{1}E_{f}+\Delta _{2}p_{f}c\right)}{\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)^{2}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}m^{2}c^{4}\sin ^{2}\Theta _{i}\left(E_{i}^{2}+E_{f}^{2}\right)-2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}p_{f}c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)+2\hbar ^{2}\omega ^{2}p_{f}m^{2}c^{3}\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\right],\\I_{4}={}&{}-{\frac {4\pi Ap_{f}c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}-{\frac {16\pi E_{i}^{2}p_{f}^{2}A\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}}{\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)^{2}}},\\I_{5}={}&{\frac {4\pi A}{\left(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\\&{}\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{f}^{2}}{E_{i}-cp_{i}\cos \Theta _{i}}}\right.\\&{}\times {\frac {E_{f}\left(2\Delta _{2}^{2}\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[\Delta _{2}^{2}+\Delta _{1}^{2}\right]\right)+p_{f}c\left(2\Delta _{1}\Delta _{2}\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]+16\Delta _{1}\Delta _{2}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}{\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&{}+{\frac {2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}\left(2\Delta _{1}\Delta _{2}p_{f}c+2\Delta _{2}^{2}E_{f}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}E_{f}\right)}{E_{i}-cp_{i}\cos \Theta _{i}}}\\&{}+{\frac {2E_{i}^{2}p_{f}^{2}\left(2\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[\left(\Delta _{1}^{2}+\Delta _{2}^{2}\right)\left(E_{f}^{2}+p_{f}^{2}c^{2}\right)+4\Delta _{1}\Delta _{2}E_{f}p_{f}c\right]\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left(E_{i}^{2}+E_{f}^{2}\right)\left(\Delta _{2}p_{f}c+\Delta _{1}E_{f}\right)}{E_{i}-cp_{i}\cos \Theta _{i}}}\right],\\I_{6}={}&{\frac {16\pi E_{f}^{2}p_{i}^{2}\sin ^{2}\Theta _{i}A}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)^{2}\left(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}},\end{aligned}}}$

and

${\displaystyle {\begin{aligned}A&={\frac {Z^{2}\alpha _{\text{fine}}^{3}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {\hbar ^{2}}{\omega }}\\\Delta _{1}&=-\mathbf {p} _{i}^{2}-\mathbf {p} _{f}^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{i}\right|\cos \Theta _{i},\\\Delta _{2}&=-2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{f}\right|+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\cos \Theta _{i}.\end{aligned}}}$

However, a much simpler expression for the same integral can be found in (Eq. 2BN) and in (Eq. 4.1).

An analysis of the doubly differential cross section above shows that electrons whose kinetic energy is larger than the rest energy (511 keV) emit photons in forward direction while electrons with a small energy emit photons isotropically.

Electron–electron bremsstrahlung

One mechanism, considered important for small atomic numbers ${\displaystyle Z}$, is the scattering of a free electron at the shell electrons of an atom or molecule. Since electron–electron bremsstrahlung is a function of ${\displaystyle Z}$ and the usual electron-nucleus bremsstrahlung is a function of ${\displaystyle Z^{2}}$, electron–electron bremsstrahlung is negligible for metals. For air, however, it plays an important role in the production of terrestrial gamma-ray flashes.

Pollution in China

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Pollution_in_China 

 

Comparison of haze and sunny days in Liaoning, China. The two images were taken 10 days apart.

Pollution in China is one aspect of the broader topic of environmental issues in China. Various forms of pollution have increased as China has industrialised, which has caused widespread environmental health problems.

Pollution statistics

The immense growth of the People's Republic of China since the 1980s has resulted in increased soil pollution. The State Environmental Protection Administration believes it to be a threat to the environment, food safety and sustainable agriculture. 38,610 square miles (100,000 km²) of China's cultivated land have been polluted, with contaminated water being used to irrigate further 31.5 million miles (21,670 km².), and another 2 million miles (1,300 km²) have been covered or destroyed by solid waste. The affected area accounts of one-tenth of China's cultivatable land. An estimated 6 million tonnes of food grain are contaminated by heavy metals every year, causing direct losses of 29 billion yuan (US$2.57 billion). Heavy metals (including mercury, lead, cadmium, copper, nickel, chromium, and zinc) in the contaminated soil have adverse health effects on human metabolism. Ingestion, contact through skin, diet through the soil-food chain, respiratory intake, and oral intake can deliver the toxic substances to human beings.

Waste

As China's waste production increases, insufficient efforts to develop capable recycling systems have been attributed to a lack of environmental awareness. In 2012, the waste generation in China was 300 million tons (229.4 kg/cap/yr).

A ban came into effect on 15 June 2008 that prohibited all supermarkets, department stores and shops throughout China from giving out free plastic bags, therefore encouraging people to use cloth bags. Stores must clearly mark the price of plastic shopping bags and are banned from adding that price onto the price of products. The production, sale and use of ultra-thin plastic bags—those less than 0.025 millimeters (0.00098  in) thick—are also banned. The State Council called for "a return to cloth bags and shopping baskets." This ban, however, does not affect the widespread use of paper shopping bags at clothing stores or the use of plastic bags at restaurants for takeout food. A survey by the International Food Packaging Association found that in the year after the ban was implemented, 10 percent fewer plastic bags found their way into the garbage.

"White pollution"

The term "white pollution" (Chinese: 白色污染; pinyin: baise wuran, less often "white garbage" Chinese: 白色垃圾; pinyin: baise laji) appears to be local to China and later to South Asia, enjoying far less use and recognition outside of the region. It refers to the color of white plastic shopping bags, styrofoam containers, and other light-colored materials that began turning up in visible volume in agricultural fields, the landscape, and waterways in the mid- to late 1990's. The first references to the term "white pollution" appear in official language at least as early as 1999, when the first bans were imposed by the State Council.

Electronic waste

Main article: Electronic waste in China

In 2011, China produced 2.3 million tons of electronic waste. The annual amount is expected to increase as the Chinese economy grows. In addition to domestic waste production, large amounts of electronic waste are imported from overseas. Legislation banning importation of electronic waste and requiring proper disposal of domestic waste has recently been introduced, but has been criticized as insufficient and susceptible to fraud. There have been local successes, such as in the city of Tianjin where 38,000 tons of electronic waste were disposed of properly in 2010, but much electronic waste is still improperly handled.

Industrial pollution

Air pollution caused by industrial plants

In 1997, the World Bank issued a report targeting China's policy towards industrial pollution. The report stated that "hundreds of thousands of premature deaths and incidents of serious respiratory illness have been caused by exposure to industrial air pollution.Since the Industrial Revolution, air pollution has been a major source of worry for human growth. Using an original survey in China, we give the first causal estimates of pollution's impact on political opinions. Seriously contaminated by industrial discharges, many of China's waterways are largely unfit for direct human use." However, the report did acknowledge that environmental regulations and industrial reforms have had some effect. It was determined that continued environmental reforms were likely to have a large effect on reducing industrial pollution.

In a 2007 article about China's pollution problem, the New York Times stated that "Environmental degradation is now so severe, with such stark domestic and international repercussions, that pollution poses not only a major long-term burden on the Chinese public but also an acute political challenge to the ruling Communist Party." The article's main points included:

1. According to the Chinese Ministry of Health, industrial pollution has made cancer China's leading cause of death.
2. Every year, ambient air pollution alone killed hundreds of thousands of citizens.
3. 500 million people in China are without safe and clean drinking water.
4. Only 1% of the country's 560 million city dwellers breathe air considered safe by the European Union, because all of its major cities are constantly covered in a "toxic gray shroud". Before and during the 2008 Summer Olympics, Beijing was "frantically searching for a magic formula, a meteorological deus ex machina, to clear its skies for the 2008 Olympics."
5. Lead poisoning or other types of local pollution continue to kill many children.
6. A large section of the ocean is without marine life because of massive algal blooms caused by the high nutrients in the water.
7. The pollution has spread internationally: sulfur dioxide and nitrogen oxides fall as acid rain on Seoul, South Korea, and Tokyo; and according to the Journal of Geophysical Research, the pollution even reaches Los Angeles in the US.
8. The Chinese Academy of Environmental Planning in 2003 produced an unpublished internal report which estimated that 300,000 people die each year from ambient air pollution, mostly of heart disease and lung cancer.
9. Chinese environmental experts in 2005 issued another report, estimating that annual premature deaths attributable to outdoor air pollution were likely to reach 380,000 in 2010 and 550,000 in 2020.
10. A 2007 World Bank report conducted with China's national environmental agency found that "[...] outdoor air pollution was already causing 350,000 to 400,000 premature deaths a year. Indoor pollution contributed to the deaths of an additional 300,000 people, while 60,000 died from diarrhoea, bladder and stomach cancer and other diseases that can be caused by water-borne pollution." World Bank officials said "China’s environmental agency insisted that the health statistics be removed from the published version of the report, citing the possible impact on 'social stability'".

A draft of a 2007 combined World Bank and SEPA report stated that up to 760,000 people died prematurely each year in China because of air and water pollution. High levels of air pollution in China's cities caused to 350,000–400,000 premature deaths. Another 300,000 died because of indoor air of poor quality. There were 60,000 premature deaths each year because of water of poor quality. Chinese officials asked that some of the results should not be published in order to avoid social unrest.

China has made some improvements in environmental protection during recent years. According to the World Bank, 'China is one of a few countries in the world that have been rapidly increasing their forest cover. It is managing to reduce air and water pollution.

Vennemo et al., in a 2009 literature review in Review of Environmental Economics and Policy, noted the wide discrepancy between the reassuring view in some Chinese official publications and the exclusively negative view in some Western sources. The review stated that "although China is starting from a point of grave pollution, it is setting priorities and making progress that resemble what occurred in industrialized countries during their earlier stages of development." Environmental trends were described as uneven. A quality of surface water in the south of China was improving and particle emissions were stable. But NO₂ emissions were increasing rapidly and SO₂ emissions had been increasing before decreasing in 2007, the last year for which data was available.

Conventional approaches to air quality monitoring are based on networks of static and sparse measurement stations. However, there are drivers behind current rises in the use of low-cost sensors for air pollution management in cities.

The immense urban growth of Chinese cities substantially increases the need for consumer goods, vehicles and energy. This in turn increases the burning of fossil fuels, resulting in smog. Exposure to Smog poses a threat to the health of Chinese citizens. A study from 2012 shows fine particles in the air, which cause respiratory and cardiovascular diseases are one of the key pollutants that are accounted for a large fraction of damage on the health of Chinese citizens.

Water pollution

The water resources of China are affected by both severe water shortages and severe water pollution.^{[citation needed]} An increasing population and rapid economic growth, as well as lax environmental oversight, have increased water demand and pollution. According to an investigation in 1980, the entire country has 440 billion cubic meters of the total water consumption. Consumption by agriculture, forestry, husbandry, and country residents was about 88 per cent of the total consumption. However, an investigation shows that 19 per cent of water in main rivers has been polluted as well as a total length of 95,000 kilometers. In addition, a survey for 878 rivers in the early 1980s shows that 80 per cent of them were polluted to some extent, and fish became extinct in more than 5 per cent of total river length throughout the country. Furthermore, there are over 20 waterways unsuitable for agricultural irrigation due to water pollution. In response, China has taken measures such as rapidly building out the water infrastructure and increased regulation as well as exploring a number of further technological solutions.

Air pollution

North-Eastern China from space, 2009. Thick haze blown off the Eastern coast of China, over Bo Hai Bay and the Yellow Sea. The haze might result from urban and industrial pollution.

In northern China, air pollution from the burning of fossil fuels, principally coal, is causing people to die on average 5.5 years sooner than they otherwise might.

Tim Flannery, Atmosphere of Hope, 2015.

Air pollution has become a major issue in China, especially in recent years, and poses a threat to Chinese public health. In 2016, only 84 out of 338 prefecture-level (administrative division of the People's Republic of China (PRC), ranking below a province and above a county) or higher cities attained the national standard for air quality. However, by 2018, those 338 cities enjoyed good air quality on 79% of days.

In the last few years, China has made significant progress in reducing air pollution. For example, average PM2.5 concentrations fell by 33% from 2013 to 2017 in 74 cities. The overall pollution in China fell further 10% between 2017 and 2018. Another study shows that China reduced PM2.5 by 47% between 2005 and 2015. In August 2019, Beijing experienced the lowest PM2.5 on record—a low of 23 micrograms per cubic meter. Beijing is on track to drop out of the Top 200 most polluted cities by the end of 2019. The reasons are many fold: (1) Millions of homes and businesses are switching from coal to natural gas and (2) Afforestation measures. China is also the world's largest producer of electric cars, but lags a number of European countries and the U.S. regarding the number of electric cars per capita.

Air pollution levels dropped in early 2020 due to quarantines addressing the coronavirus pandemic. By early 2021, however, the levels had risen again.

The Chinese government realized that the pollution had an effect on its regime's satisfaction. So therefore, the Chinese government spent a lot of money to combat pollution.An example of this is that in 2013, China's Academy for Environmental Planning pledged $277 billion to combat urban air pollution. In the first batch of 74 cities that implemented the 2012 Environmental Air Quality Standards, the average concentration of PM2.5 and sulfur dioxide dropped by 42 percent and 68 percent, respectively, between 2013 and 2018.

Zhong Nanshan, the president of the China Medical Association, warned in 2012 that air pollution could become China's biggest health threat. Measurements by Beijing municipal government in January 2013 showed that highest recorded level of PM2.5 (particulate matter smaller than 2.5 micrometers in size), was at nearly 1,000 μg per cubic meter. PM_2.5, consisting of K⁺, Ca²⁺, NO₃⁻, and SO₄^2-, had the most fearsome impact on people's health in Beijing throughout the year, especially in cold seasons. Traces of smog from mainland China has been observed to reach as far as California.

Sulfur dioxide emission peaked at 2006, after which it began to decline by 10.4% in 2008 compared to 2006. This was accompanied by improvements on related phenomenons such as lower frequency of acid rainfall. The adoption by power plants of flue-gas desulfurization technology was likely the main reason for reduced SO₂ emissions.

Large-scale use of formaldehyde in make home building products in construction and furniture also contributes to indoor air pollution.

Particulates

Particulates are formed from both primary and secondary pathways. Primary sources such as coal combustion, biomass combustion and traffic directly emit particulate matter (PM). The emissions from power plants are considerably higher than in other countries, as most Chinese facilities do not employ any flue gas treatment. High secondary aerosol (particulates formed through atmospheric oxidation and reactions of gaseous organic compounds) contribution to particulate pollution in China is found. According to the U.S. Environmental Protection Agency, such fine particles can cause asthma, bronchitis, and acute and chronic respiratory symptoms such as shortness of breath and painful breathing, and may also lead to premature death.

According to the World Bank, the Chinese cities with the highest levels of particulate matter in 2004 of those studied were Tianjin, Chongqing, and Shenyang. In 2012 stricter air pollution monitoring of ozone and PM2.5 were ordered to be gradually implemented from large cities and key areas to all prefecture-level cities, and from 2015 all prefecture-level or higher cities were included. State media acknowledged the role of environmental campaigners in causing this change. On one micro-blog service, more than a million mostly positive comments were posted in less than 24 hours although some wondered if the standards would be effectively enforced.

The US embassy in Beijing regularly posts automated air quality measurements at @beijingair on Twitter. On 18 November 2010, the feed described the PM2.5 AQI (Air Quality Index) as "crazy bad" after registering a reading in excess of 500 for the first time. This description was later changed to "beyond index", a level which recurred in February, October, and December 2011.

In June 2012, following strongly divergent disclosures of particulate levels between the Observatory and the US Embassy, Chinese authorities asked foreign consulates to stop publishing "inaccurate and unlawful" data. Officials said it was "not scientific to evaluate the air quality of an area with results gathered from just only one point inside that area", and asserted that official daily average PM2.5 figures for Beijing and Shanghai were "almost the same with the results published by foreign embassies and consulates".

By January 2013 the pollution had worsened with official Beijing data showing an average AQI over 300 and readings of up to 700 at individual recording stations while the US Embassy recorded over 755 on 1 January and 800 by 12 January 2013.

On 21 October 2013, record smog closed the Harbin Airport along with all schools in the area. Daily particulate levels of more than 50 times the World Health Organization recommended daily level were reported in parts of the municipality.

In 2016, Beijing's yearly-average PM2.5 was 73 μg/m³, 9.9% improvement compared to 2015. In total, 39 severely polluted days were recorded, 5 fewer compared to 2015.

2016 Air pollution in Beijing as measured by Air Quality Index (AQI)

   Severely polluted

   Heavily polluted

   Moderately polluted

   Lightly polluted

   Good

  Excellent

Government's response to the air pollution

In an attempt to reduce air pollution, the Chinese government has made the decision to enforce stricter regulations. After record-high air pollution in northern China in 2012 and 2013, the State Council issued an Action Plan for the Prevention and Control of Air Pollution in September 2013. This plan aims to reduce PM2.5 by over 10% from 2012 to 2017. The most prominent government response has been in Beijing, aiming to reduce PM2.5 by 25% from 2012 to 2017. As the capital of China, it is suffering from high levels of air pollution. According to Reuters, in September 2013, the Chinese government published the plan to tackle air pollution problem on its official website. The main goal of the plan is to reduce coal consumption by closing polluting mills, factories, and smelters, and switching to other eco-friendly energy sources.

These policies have been taking effect, and in 2015, the average PM2.5 in 74 key cities in monitoring system is 55 μg/m³, showing a 23.6% decrease as of 2013. Despite the reduction in coal consumption and polluting industries, China still maintained a stable economic growth rate from 7.7% in 2013 to 6.9% in 2015.

On 20 August 2015, ahead of the 70th-anniversary celebrations of the end of World War II, the Beijing government shut down industrial facilities and reduced car emissions in order to achieve a "Parade Blue" sky for the occasion. This action resulted in PM2.5 concentration lower than the 35 μg/m³ national air quality standard, according to data from Beijing Municipal Environmental Protection Monitoring Centre (BMEMC). The restrictions resulted in an average Beijing PM2.5 concentration of 19.5 μg/m³, the lowest that had ever been on record in the capital.

China's strategy has been mainly focusing on the development of other energy sources such as nuclear, hydro and compressed natural gas. The latest plan entails closing the outdated capacity of the industrial sectors like iron, steel, aluminum and cement and increasing nuclear capacity and other non-fossil fuel energy. It also includes an intention to stop approving new thermal power plants and to cut coal consumption in industrial areas.

According to research, substituting all coal consumption for residential and commercial use to natural gas requires additional 88 billion cubic meters of natural gas, which is 60% of China's total consumption in 2012, and the net cost would be 32–52 billion dollars. Substituting the share of coal-fired power plant with renewable and nuclear energy also requires 700GW additional capacity, which cost 184 billion dollars. So the net cost would be 140–160 billion dollars considering value of saved coal. Since all the above policies have been already partially implemented by national and city governments, they should lead to substantial improvements in urban air quality.

In northern China, air pollution from the burning of fossil fuels, principal coal, is causing people to die on average 5.5 years sooner than they otherwise might.

— Tim Flannery, Atmosphere of Hope, 2015.

Four-color alert system

Beijing launched four-color alert system in 2013. It is based on the air quality index (AQI), which indicates how clean or polluted the air is.

The Beijing government revised their four-color alert system at the start of 2016, increasing the levels of pollution required to trigger orange and red alerts. The change was introduced to standardize the alert levels across four cities including Tianjin and four cities in Hebei, and perhaps in direct response to the red alerts issues the previous December.

AQI	Description
101–150	Slight pollution
151–200	Moderate pollution
201–300	Heavy pollution
301–500	Hazardous

Color	Condition
Blue	"Heavy pollution" in the next 24 hours
Yellow	"Hazardous" in the next 24 hours; or "heavy pollution" for three consecutive days
Orange	Alternate "heavy pollution" and "hazardous" days for three consecutive days
Red	Average of "heavy pollution" for four consecutive days, "hazardous" for two consecutive days or average AQI over 500 for one day

Light pollution

With active economic growth and a huge number of citizens, China is considered as the largest developing country in the world. Due to urbanization, light pollution generally is an environmental factor that significantly influences the quality and health of wildlife. According to Pengpeng Han et al., "In the 1990s, the increasing trend in light pollution regions mostly occurred in larger urban cities, which are mainly located in eastern and coastal areas, whereas the decreasing trend areas were chiefly industrial and mining cities rich in mineral resources, in addition to the central parts of large cities". In the 2000s, nearly all urban cities were dominated by an uprising trend in light pollution.

Common pollutants

Lead

Lead poisoning was described in a 2001 paper as one of the most common pediatric health problems in China. A 2006 review of existing data suggested that one-third of Chinese children suffer from elevated serum lead levels. Pollution from metal smelters and a fast-growing battery industry has been responsible for most cases of, particularly high lead levels. In 2011, there were riots in the Zhejiang Haijiu Battery Factory from angry parents whose children received permanent neurological damage from lead poisoning. The central government has acknowledged the problem and has taken measures such as suspending battery factory production, but some see the response as inadequate and some local authorities have tried to silence criticisms.

A literature review of academic studies on Chinese children's blood lead levels found that the lead levels declined when comparing the studies published during 1995–2003 and 2004–2007 periods. Lead levels also showed a declining trend after China banned lead in gasoline in 2000. Lead levels were still higher than those in developed nations. Industrial areas had higher levels than suburban areas, which had higher levels than urban areas. Controlling and preventing lead poisoning was described as a long-term mission.

Persistent organic pollutants

China is a signatory nation of the Stockholm Convention, a treaty to control and phase out major persistent organic pollutants (POP). A plan of action for 2010 includes objectives such as eliminating the production, import and use of the pesticides covered under the convention, as well as an accounting system for PCB containing equipment. For 2015, China plans to establish an inventory of POP-contaminated sites and remediation plans. Since May 2009, this treaty also covers polybrominated diphenyl ethers and perfluorooctanesulfonic acid. Perfluorinated compounds are associated with altered thyroid function and decreased sperm count in humans. China faces a big challenge in controlling and eliminating POPs, since they often are cheaper than their alternatives, or are unintentionally produced and then released into the environment to save on treatment costs.

Yellow dust

The Yellow dust or Asian dust is a seasonal dust cloud which affects Northeast Asia during late winter and springtime. The dust originates in the deserts of Mongolia, northern China and Kazakhstan where high-speed surface winds and intense dust storms kick up dense clouds of fine, dry soil particles. These clouds are then carried eastward by prevailing winds and pass over Northern China into Korea and Japan.

Desertification has intensified in China. 1,740,000 square kilometres of land is classified as "dry", and desertification disrupts the lives of 400 million people and causes direct economic losses of 54 billion yuan ($7 billion) a year, SFA figures show. Sulfur (an acid rain component), soot, ash, carbon monoxide, and other toxic pollutants including heavy metals (such as mercury, cadmium, chromium, arsenic, lead, zinc, copper) and other carcinogens, often accompany the dust storms, as well as viruses, bacteria, fungi, pesticides, antibiotics, asbestos, herbicides, plastic ingredients, combustion products and hormone mimicking phthalates.

Coal

The increasing number of air pollutants can cause incidences of low visibility for days and acid rain. According to the article "Air Pollution in Mega Cities in China", "Coal accounts for 70% of the total energy consumption, and emissions from coal combustion are the major anthropogenic contributors to air pollution in China." The Proceedings of the National Academy of Sciences also highlights the Huai River Policy established during China's central planning period between 1950 and 1980. The policy provided homes and offices with free coal for winter heating but was limited solely to the Northern region due to budget limitations. The policy led to a dramatic increase in coal consumption and production. Coal production alongside rapid economic growth has increased the emission of harmful pollutants such as carbon dioxide, sulfur dioxide, nitrogen oxide, and small particle matter known as PM2.5 and PM10. Long-term exposure to pollutants can cause health risks such as respiratory diseases, cancer, cardiovascular and cerebrovascular diseases. Coal is a huge issue because of the SO2 emissions from coal factories. According to the article, "SO2 exceeded the Chinese Grade-II standards in 22% of the country’s cities and caused acid rain problems in 38% of the cities."

Other pollutants

In 2010 49 employees at Wintek were poisoned by n-hexane in the manufacturing of touchscreens for Apple products.

In 2013, it was revealed that portions of the country's rice supply were tainted with the toxic metal cadmium.

Impact of pollution

Smog in Beijing, 2013

A 2006 Chinese green gross domestic product estimate stated that pollution in 2004 cost 3.05% of the nation's economy.

A 2007 World Bank and SEPA report estimated the cost of water and air pollution in 2003 to 2.68% or 5.78% of GDP depending on if using a Chinese or a Western method of calculation.

A 2009 review stated a range of 2.2–10% of GDP.

A 2012 study stated that pollution had little effect on economic growth which in China's case was largely dependent on physical capital expansion and increased energy consumption due to the dependency on manufacturing and heavy industries. China was predicted to continue to grow using energy-inefficient and polluting industries. While growth may continue, the rewards of this growth may be opposed by the harm from the pollution unless environmental protection is increased.

A 2013 study published in the Proceedings of the National Academy of Sciences found that severe pollution during the 1990s cut five and a half (5.5) years from the average life expectancy of people living in northern China, where toxic air has led to increased rates of stroke, heart disease and cancer.

A 2015 study from the non-profit organization Berkeley Earth estimated that 1.6 million people in China die each year from heart, lung and stroke problems because of polluted air.

Cross-border pollution

See also: Air pollution in Taiwan

Criticisms of government environmental policies

Global carbon dioxide emissions by jurisdiction.

Critics point to the government's lack of willingness to protect the environment as a common problem with China's environmental policies. Even in the case of the latest plan, experts are skeptical about its actual influence because of the existence of loopholes. This is because economic growth is still the primary issue for the government, and overrides environmental protection.

However, if the measures to cut coal usage were applied strictly, it would also mean the dismantling of the local economy that is highly reliant on heavy industry. The Financial Times interviewed a worker who stated, "if this steel mill didn’t exist, we wouldn’t even have anywhere to go to eat. Everything revolves around this steel factory – our children work here."

Scientists have yet to agree on the impact of China's air pollution on neighboring countries. Some politicians in South Korea claim that more than half of Korea's air pollution is caused by fine dust generated in China, but China disagrees.

Pollution ratings

As of 2019:

• The top five cities with the best air quality: Lhasa, Haikou, Zhoushan, Xiamen, Huangshan
• The 10 cities with the worst air quality: Anyang, Xingtai, Shijiazhuang, Handan, Linfen, Tangshan, Taiyuan, Zibo, Jiaozuo, Jincheng

According to the National Environmental Analysis released by Tsinghua University and The Asian Development Bank in January 2013, seven of the ten most air polluted cities in the world are in China, including Taiyuan, Beijing, Urumqi, Lanzhou, Chongqing, Jinan and Shijiazhuang.

National Sword Policy

The Operation National Sword was a policy initiative launched in 2017 by the Government of China to monitor and more stringently review recyclable waste imports. Before the policy, China was importing the vast majority of recyclables from North America and Europe for two decades. This practice of buying recyclables brought raw materials for the growing industrial capacity of China, but also brought a lot of contaminated recyclables which ended up accruing in China, causing other environmental concerns such as air and water pollution.

The action was interpreted as an international relations move by China against Western countries. The policy caused a ripple effect in the global recyclables market, causing major pile ups in Western countries who had been collecting lower quality recyclables in single-stream recycling, and displacing some of those recyclable to other countries, mostly in South East Asia, like Vietnam and Malaysia.

Proof of impossibility

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Proof_of_impossibility 

In mathematics, a proof of impossibility, also known as negative proof, proof of an impossibility theorem, or negative result is a proof demonstrating that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Proofs of impossibility often put decades or centuries of work attempting to find a solution to rest. To prove that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a theory. Impossibility theorems are usually expressible as negative existential propositions, or universal propositions in logic (see universal quantification for more).

Perhaps one of the oldest proofs of impossibility is that of the irrationality of square root of 2, which shows that it is impossible to express the square root of 2 as a ratio of integers. Another famous proof of impossibility was the 1882 proof of Ferdinand von Lindemann, showing that the ancient problem of squaring the circle cannot be solved, because the number π is transcendental (i.e., non-algebraic) and only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century.

A problem arising in the 16th century was that of creating a general formula using radicals expressing the solution of any polynomial equation of fixed degree k, where k ≥ 5. In the 1820s, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) showed this to be impossible, using concepts such as solvable groups from Galois theory—a new subfield of abstract algebra.

Among the most important proofs of impossibility of the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm at all, with the most famous one being the halting problem. Other similarly-related findings are those of the Gödel's incompleteness theorems, which uncovers some fundamental limitations in the provability of formal systems.

In computational complexity theory, techniques like relativization (see oracle machine) provide "weak" proofs of impossibility excluding certain proof techniques. Other techniques, such as proofs of completeness for a complexity class, provide evidence for the difficulty of problems, by showing them to be just as hard to solve as other known problems that have proved intractable.

Types of impossibility proof

See also: Types of proof

Proof by contradiction

One widely used type of impossibility proof is proof by contradiction. In this type of proof, it is shown that if something, such as a solution to a particular class of equations, were possible, then two mutually contradictory things would be true, such as a number being both even and odd. The contradiction implies that the original premise is impossible.

Proof by descent

Main article: Proof by infinite descent

One type of proof by contradiction is proof by descent, which proceeds first by assuming that something is possible, such as a positive integer solution to a class of equations, and that therefore there must be a smallest solution. From the alleged smallest solution, it is then shown that a smaller solution can be found, contradicting the premise that the former solution was the smallest one possible—thereby showing that the original premise (that a solution exists) must be false.

Types of disproof of impossibility conjectures

There are two alternative methods of disproving a conjecture that something is impossible: by counterexample (constructive proof) and by logical contradiction (non-constructive proof).

The obvious way to disprove an impossibility conjecture by providing a single counterexample. For example, Euler proposed that at least n different n^th powers were necessary to sum to yet another n^th power. The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power:

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵.

A proof by counterexample is a constructive proof, in that an object disproving the claim is exhibited. In contrast, a non-constructive proof of an impossibility claim would proceed by showing it is logically contradictory for all possible counterexamples to be invalid: At least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility conjecture. For example, a conjecture that it is impossible for an irrational power raised to an irrational power to be rational was disproved, by showing that one of two possible counterexamples must be a valid counterexample, without showing which one it is.

The existence of irrational numbers: The Pythagoreans' proof

The proof by Pythagoras (or more likely one of his students) about 500 BCE has had a profound effect on mathematics. It shows that the square root of 2 cannot be expressed as the ratio of two integers (counting numbers). The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers. This bifurcation was used by Cantor in his diagonal method, which in turn was used by Turing in his proof that the Entscheidungsproblem, the decision problem of Hilbert, is undecidable.

It is unknown when, or by whom, the "theorem of Pythagoras" was discovered. The discovery can hardly have been made by Pythagoras himself, but it was certainly made in his school. Pythagoras lived about 570–490 BCE. Democritus, born about 470 BCE, wrote on irrational lines and solids ...

— Heath

Proofs followed for various square roots of the primes up to 17.

There is a famous passage in Plato's Theaetetus in which it is stated that Teodorus (Plato's teacher) proved the irrationality of

${\displaystyle {\sqrt {3}},{\sqrt {5}},...,}$

taking all the separate cases up to the root of 17 square feet ... .

A more general proof now exists that:

The mth root of an integer N is irrational, unless N is the mth power of an integer n".

That is, it is impossible to express the mth root of an integer N as the ratio a⁄b of two integers a and b, that share no common prime factor except in cases in which b = 1.

Impossible constructions sought by the ancient Greeks

Three famous questions of Greek geometry were how:

1. ... with compass and straight-edge to trisect any angle,
2. to construct a cube with a volume twice the volume of a given cube
3. to construct a square equal in area to that of a given circle.

For more than 2,000 years unsuccessful attempts were made to solve these problems; at last, in the 19th century it was proved that the desired constructions are logically impossible.

A fourth problem of the ancient Greeks was to construct an equilateral polygon with a specified number n of sides, beyond the basic cases n = 3, 4, 5, 6 that they knew how to construct.

All of these are problems in Euclidean construction, and Euclidean constructions can be done only if they involve only Euclidean numbers (by definition of the latter) (Hardy and Wright p. 159). Irrational numbers can be Euclidean. A good example is the irrational number the square root of 2. It is simply the length of the hypotenuse of a right triangle with legs both one unit in length, and it can be constructed with straightedge and compass. But it was proved centuries after Euclid that Euclidean numbers cannot involve any operations other than addition, subtraction, multiplication, division, and the extraction of square roots.

Angle trisection and doubling the cube

Both trisecting the general angle and doubling the cube require taking cube roots, which are not constructible numbers by compass and straightedge.

Squaring the circle

${\displaystyle \pi }$ is not a Euclidean number ... and therefore it is impossible to construct, by Euclidean methods a length equal to the circumference of a circle of unit diameter

A proof exists to demonstrate that any Euclidean number is an algebraic number—a number that is the solution to some polynomial equation. Therefore, because ${\displaystyle \pi }$ was proved in 1882 to be a transcendental number and thus by definition not an algebraic number, it is not a Euclidean number. Hence the construction of a length ${\displaystyle \pi }$ from a unit circle is impossible, and the circle cannot be squared.

Constructing an equilateral n-gon

The Gauss-Wantzel theorem showed in 1837 that constructing an equilateral n-gon is impossible for most values of n.

Euclid's parallel axiom

Main article: Non-Euclidean geometry

Nagel and Newman consider the question raised by the parallel postulate to be "...perhaps the most significant development in its long-range effects upon subsequent mathematical history" (p. 9).

The question is: can the axiom that two parallel lines "...will not meet even 'at infinity'" (footnote, ibid) be derived from the other axioms of Euclid's geometry? It was not until work in the nineteenth century by "... Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. ...a proof can be given of the impossibility of proving certain propositions [in this case, the parallel postlate] within a given system [in this case, Euclid's first four postulates]". (p. 10)

Fermat's Last Theorem

Main article: Fermat's Last Theorem

Fermat's Last Theorem was conjectured by Pierre de Fermat in the 1600s, states the impossibility of finding solutions in positive integers for the equation ${\displaystyle x^{n}+y^{n}=z^{n}}$ with ${\displaystyle n>2}$. Fermat himself gave a proof for the n = 4 case using his technique of infinite descent, and other special cases were subsequently proved, but the general case was not proved until 1994 by Andrew Wiles.

Richard's paradox

Main article: Richard's paradox

This profound paradox presented by Jules Richard in 1905 informed the work of Kurt Gödel (cf Nagel and Newman p. 60ff) and Alan Turing. A succinct definition is found in Principia Mathematica:

Richard's paradox ... is as follows. Consider all decimals that can be defined by means of a finite number of words [“words” are symbols; boldface added for emphasis]; let E be the class of such decimals. Then E has ${\displaystyle \aleph _{0}}$ [an infinite number of] terms; hence its members can be ordered as the 1st, 2nd, 3rd, ... Let X be a number defined as follows [Whitehead & Russell now employ the Cantor diagonal method].
If the n-th figure in the n-th decimal is p, let the n-th figure in X be p + 1 (or 0, if p = 9). Then X is different from all the members of E, since, whatever finite value n may have, the n-th figure in X is different from the n-th figure in the n-th of the decimals composing E, and therefore X is different from the n-th decimal. Nevertheless we have defined X in a finite number of words [i.e. this very definition of “word” above.] and therefore X ought to be a member of E. Thus X both is and is not a member of E.

— Principia Mathematica, 2nd edition 1927, p. 61

Kurt Gödel considered his proof to be “an analogy” of Richard's paradox, which he called “Richard's antinomy”. See more below about Gödel's proof.

Alan Turing constructed this paradox with a machine and proved that this machine could not answer a simple question: will this machine be able to determine if any machine (including itself) will become trapped in an unproductive ‘infinite loop’ (i.e. it fails to continue its computation of the diagonal number).

Can this theorem be proved from these axioms? Gödel's proof

Main article: Gödel's incompleteness theorems

To quote Nagel and Newman (p. 68), "Gödel's paper is difficult. Forty-six preliminary definitions, together with several important preliminary theorems, must be mastered before the main results are reached" (p. 68). In fact, Nagel and Newman required a 67-page introduction to their exposition of the proof. But if the reader feels strong enough to tackle the paper, Martin Davis observes that "This remarkable paper is not only an intellectual landmark, but is written with a clarity and vigor that makes it a pleasure to read" (Davis in Undecidable, p. 4). It is recommended that most readers see Nagel and Newman first.

So what did Gödel prove? In his own words:

"It is reasonable... to make the conjecture that ...[the] axioms [from Principia Mathematica and Peano ] are ... sufficient to decide all mathematical questions which can be formally expressed in the given systems. In what follows it will be shown that this is not the case, but rather that ... there exist relatively simple problems of the theory of ordinary whole numbers which cannot be decided on the basis of the axioms" (Gödel in Undecidable, p. 4).

Gödel compared his proof to "Richard's antinomy" (an "antinomy" is a contradiction or a paradox; for more see Richard's paradox):

"The analogy of this result with Richard's antinomy is immediately evident; there is also a close relationship [14] with the Liar Paradox (Gödel's footnote 14: Every epistemological antinomy can be used for a similar proof of undecidability)... Thus we have a proposition before us which asserts its own unprovability [15]. (His footnote 15: Contrary to appearances, such a proposition is not circular, for, to begin with, it asserts the unprovability of a quite definite formula)" (Gödel in Undecidable, p.9).

Will this computing machine lock in a "circle"? Turing's first proof

Main article: Turing's proof

• The Entscheidungsproblem, the decision problem, was first answered by Church in April 1935 and preempted Turing by over a year, as Turing's paper was received for publication in May 1936. (Also received for publication in 1936—in October, later than Turing's—was a short paper by Emil Post that discussed the reduction of an algorithm to a simple machine-like "method" very similar to Turing's computing machine model (see Post–Turing machine for details).
• Turing's proof is made difficult by number of definitions required and its subtle nature. See Turing machine and Turing's proof for details.
• Turing's first proof (of three) follows the schema of Richard's paradox: Turing's computing machine is an algorithm represented by a string of seven letters in a "computing machine". Its "computation" is to test all computing machines (including itself) for "circles", and form a diagonal number from the computations of the non-circular or "successful" computing machines. It does this, starting in sequence from 1, by converting the numbers (base 8) into strings of seven letters to test. When it arrives at its own number, it creates its own letter-string. It decides it is the letter-string of a successful machine, but when it tries to do this machine's (its own) computation it locks in a circle and can't continue. Thus we have arrived at Richard's paradox. (If you are bewildered see Turing's proof for more).

A number of similar undecidability proofs appeared soon before and after Turing's proof:

1. April 1935: Proof of Alonzo Church ("An Unsolvable Problem of Elementary Number Theory"). His proof was to "...propose a definition of effective calculability ... and to show, by means of an example, that not every problem of this class is solvable" (Undecidable p. 90))
2. 1946: Post correspondence problem (cf Hopcroft and Ullman p. 193ff, p. 407 for the reference)
3. April 1947: Proof of Emil Post (Recursive Unsolvability of a Problem of Thue) (Undecidable p. 293). This has since become known as "The Word problem of Thue" or "Thue's Word Problem" (Axel Thue proposed this problem in a paper of 1914 (cf References to Post's paper in Undecidable, p. 303)).
4. Rice's theorem: a generalized formulation of Turing's second theorem (cf Hopcroft and Ullman p. 185ff)
5. Greibach's theorem: undecidability in language theory (cf Hopcroft and Ullman p. 205ff and reference on p. 401 ibid: Greibach [1963] "The undecidability of the ambiguity problem for minimal lineal grammars," Information and Control 6:2, 117–125, also reference on p. 402 ibid: Greibach [1968] "A note on undecidable properties of formal languages", Math Systems Theory 2:1, 1–6.)
6. Penrose tiling questions
7. Question of solutions for Diophantine equations and the resultant answer in the MRDP Theorem; see entry below.

Can this string be compressed? Chaitin's proof

Main article: Chaitin's incompleteness theorem

For an exposition suitable for non-specialists see Beltrami p. 108ff. Also see Franzen Chapter 8 pp. 137–148, and Davis pp. 263–266. Franzén's discussion is significantly more complicated than Beltrami's and delves into Ω—Gregory Chaitin's so-called "halting probability". Davis's older treatment approaches the question from a Turing machine viewpoint. Chaitin has written a number of books about his endeavors and the subsequent philosophic and mathematical fallout from them.

A string is called (algorithmically) random if it cannot be produced from any shorter computer program. While most strings are random, no particular one can be proved so, except for finitely many short ones:

"A paraphrase of Chaitin's result is that there can be no formal proof that a sufficiently long string is random..." (Beltrami p. 109)

Beltrami observes that "Chaitin's proof is related to a paradox posed by Oxford librarian G. Berry early in the twentieth century that asks for 'the smallest positive integer that cannot be defined by an English sentence with fewer than 1000 characters.' Evidently, the shortest definition of this number must have at least 1000 characters. However, the sentence within quotation marks, which is itself a definition of the alleged number is less than 1000 characters in length!" (Beltrami, p. 108)

Does this Diophantine equation have an integer solution? Hilbert's tenth problem

Main articles: Matiyasevich's theorem and Hilbert's problems

The question "Does any arbitrary "Diophantine equation" have an integer solution?" is undecidable.That is, it is impossible to answer the question for all cases.

Franzén introduces Hilbert's tenth problem and the MRDP theorem (Matiyasevich-Robinson-Davis-Putnam theorem) which states that "no algorithm exists which can decide whether or not a Diophantine equation has any solution at all". MRDP uses the undecidability proof of Turing: "... the set of solvable Diophantine equations is an example of a computably enumerable but not decidable set, and the set of unsolvable Diophantine equations is not computably enumerable" (p. 71).

In social science

In political science, Arrow's impossibility theorem states that it is impossible to devise a voting system that satisfies a set of five specific axioms. This theorem is proved by showing that four of the axioms together imply the opposite of the fifth.

In economics, Holmström's theorem is an impossibility theorem proving that no incentive system for a team of agents can satisfy all of three desirable criteria.

In natural science

Main article: No-go theorem

In natural science, impossibility assertions (like other assertions) come to be widely accepted as overwhelmingly probable rather than considered proved to the point of being unchallengeable. The basis for this strong acceptance is a combination of extensive evidence of something not occurring, combined with an underlying theory, very successful in making predictions, whose assumptions lead logically to the conclusion that something is impossible.

Two examples of widely accepted impossibilities in physics are perpetual motion machines, which violate the law of conservation of energy, and exceeding the speed of light, which violates the implications of special relativity. Another is the uncertainty principle of quantum mechanics, which asserts the impossibility of simultaneously knowing both the position and the momentum of a particle. Also Bell's theorem: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

While an impossibility assertion in natural science can never be absolutely proved, it could be refuted by the observation of a single counterexample. Such a counterexample would require that the assumptions underlying the theory that implied the impossibility be re-examined.