Lorentz force

Path of an electron of velocity v moving in a magnetic field B. Where the dotted circle indicates the magnetic field directed out of the plane, and the crossed circle indicates the magnetic field directed into the plane.

The Lorentz force, named after Dutch physicist Hendrik Lorentz, for a charged particle q is given (in SI units) by the equation:^[13]

${\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} }$

where qE describes the electric force due to a present electric field,E, acting on q.
And qv x B describes the magnetic force due to a present magnetic field, B, acting on q when q is moving with velocity v.^[13][14]
Which can also be written as:

${\displaystyle \mathbf {F} =q[-\nabla \phi -{\frac {d\mathbf {A} }{dt}}+\nabla (\mathbf {A} \cdot \mathbf {v} )]}$

where ${\displaystyle \phi }$ is the electric potential, and A is the magnetic vector potential.^[15]

It was Oliver Heaviside who is attributed in 1885 and 1889 to first deriving the correct expression for the Lorentz force of qv x B.^[16] Hendrik Lorentz derived and refined the concept in 1892 and gave it his name,^[17] incorporating forces due to electric fields.
Rewriting this as the equation of motion for a free particle of charge q mass m,this becomes:^[13]

${\displaystyle m{\frac {d\mathbf {v} }{dt}}=q\mathbf {E} +q\mathbf {v} \times \mathbf {B} }$

or

${\displaystyle m{\frac {d\gamma \mathbf {v} }{dt}}=q\mathbf {E} +q\mathbf {v} \times \mathbf {B} }$

in the relativistic case using Lorentz contraction where γ is:^[18]

${\displaystyle \gamma (v)\equiv {\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$

this equation of motion was first verified in 1897 in J. J. Thomson's experiment investigating cathode rays which confirmed, through bending of the rays in a magnetic field, that these rays were a stream of charged particles now known as electrons.^[10][13]

Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a particle which might be traveling near the speed of light (relativistic form of the Lorentz force).

Electrostatic Coulomb force

The absolute value of the force F between two point charges q and Q relates to the distance r between the point charges and to the simple product of their charges. The diagram shows that like charges repel each other, and opposite charges attract each other.

In the image, the vector F₁ is the force experienced by q₁, and the vector F₂ is the force experienced by q₂. When q₁q₂ > 0 the forces are repulsive (as in the image) and when q₁q₂ < 0 the forces are attractive (opposite to the image). The magnitude of the forces will always be equal. In this case: ${\displaystyle \mathbf {F} =k{\frac {q_{1}q_{2}}{|\mathbf {r_{12}} |^{2}}}\mathbf {\hat {r_{12}}} ={\frac {q_{1}q_{2}}{4\pi \epsilon _{0}|\mathbf {r_{12}} |^{2}}}\mathbf {\hat {r_{12}}} }$
where the vector,
${\displaystyle {\boldsymbol {r_{12}}}={\boldsymbol {r_{1}-r_{2}}}}$
is the vectorial distance between the charges and, ${\displaystyle {\boldsymbol {{\hat {r}}_{12}}}={{\boldsymbol {r_{12}}}/|{\boldsymbol {r_{12}}}|}}$
(a unit vector pointing from q₂ to q₁).
The vector form of the equation above calculates the force F₁ applied on q₁ by q₂. If r₂₁ is used instead, then the effect on q₂ can be found. It can be also calculated using Newton's third law: F₂ = -F₁.

Electrostatic Coulomb force also known as Coulomb interaction and electrostatic force, named for Charles-Augustin de Coulomb who published the result in 1785, describes the attraction or repulsion of particles due to their electric charge.^[19]

Coulomb's law states that:

The magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.^{[20][note 5]}

The magnitude of the electrostatic force is proportional to the scalar multiple of the charge magnitudes, and inversely proportional to the square of the distance (i.e. Inverse-square law), and is given by:

${\displaystyle F=k{\frac {|q_{1}q_{2}|}{r^{2}}}={\frac {|q_{1}q_{2}|}{4\pi \epsilon _{0}r^{2}}}}$

or in vector notation:

${\displaystyle \mathbf {F} =k{\frac {q_{1}q_{2}}{|\mathbf {r} |^{2}}}\mathbf {\hat {r}} ={\frac {q_{1}q_{2}}{4\pi \epsilon _{0}|\mathbf {r} |^{2}}}\mathbf {\hat {r}} }$

where q₁,q₂ are two signed point charges; r-hat being the unit vector direction of the distance r between charges; k is Coulombs constant and ε₀ is the permittivity of free space, given in SI units by:^[20]

${\displaystyle k={\frac {1}{4\pi \epsilon _{0}}}\approx 8.988\times 10^{9}Nm^{2}{C^{-}}^{2}}$

${\displaystyle \epsilon _{0}\approx 8.854\times 10^{-12}C^{2}N^{-1}m^{-2}}$

The directions of the forces exerted by the two charges on one another are always along the straight line joining them (the shortest distance), and are vector forces of infinite range; and obey Newtons 3rd law being of equal magnitude and opposite direction. Further, when both charges q₁ and q₂ have the same sign (either both positive or both negative) the forces between them are repulsive, if they are of opposite sign then the forces are attractive.^[20][21] These forces obey an important property called the principle of superposition of forces which states that if a third charge were introduced then the total force acting on that charge is the vector sum of the forces that would be exerted by the other charges individually, this holds for any number of charges.^[20] However, Coulomb's Law has been stated for charges in a vacuum, if the space between point charges contains matter then the permittivity of the matter between the charges must be accounted for as follows:

${\displaystyle F=k{\frac {|q_{1}q_{2}|}{\epsilon _{r}r^{2}}}}$

where ε_r is the relative permittivity or dielectric constant of the space the force acts through, and is dimensionless.^[20]

Collisions

If two particles interact with one another in a scattering process there are two results possible after the interaction:^[22]

Electron-molecule scattering

Electron scattering by isolated atoms and molecules occurs in the gas phase. It plays a key role in plasma physics and chemistry and it's important for such applications as semiconductor physics. Electron-molecule/atom scattering is normally treated by means of quantum mechanics. The leading approach to compute the cross sections is using R-matrix method.

Compton scattering

Compton Scattering Feynman Diagram

Compton scattering, so named for Arthur Compton who first observed the effect in 1922 and which earned him the 1927 Nobel Prize in Physics;^[25] is the inelastic scattering of a high-energy photon by a free charged particle.^{[26][note 6]}

This was demonstrated in 1923 by firing radiation of a given wavelength (X-rays in the given case) through a foil (carbon target), which was scattered in a manner inconsistent with classical radiation theory.^{[26][note 7]} Compton published a paper in the Physical Review explaining the phenomenon: A quantum theory of the scattering of X-rays by light elements.^[27] The Compton effect can be understood as high-energy photons scattering in-elastically off individual electrons,^[26] when the incoming photon gives part of its energy to the electron, then the scattered photon has lower energy and lower frequency and longer wavelength according to the Planck relation:^[28]

${\displaystyle E=h\nu =hf}$

which gives the energy E of the photon in terms of frequency f or ν, and Planck's constant h (6.626×10⁻³⁴ J⋅s = 4.136×10⁻¹⁵ eV.s).^[29] The wavelength change in such scattering depends only upon the angle of scattering for a given target particle.^[28][30]

This was an important discovery during the 1920s when the particle (photon) nature of light suggested by the photoelectric effect was still being debated, the Compton experiment gave clear and independent evidence of particle-like behavior.^[25][30]

The formula describing the Compton shift in the wavelength due to scattering is given by:

${\displaystyle \lambda _{f}-\lambda _{i}={\frac {h}{m_{e}c}}(1-\cos \theta )}$

where λ_f is the final wavelength of the photon after scattering, λ_i is the initial wavelength of the photon before scattering, h is Planck's constant, m_e is the rest mass of the electron, c is the speed of light and θ is the scattering angle of the photon.^[25][30]

The coefficient of (1 - cos θ) is known as the Compton wavelength, but is in fact a proportionality constant for the wavelength shift.^[31] The collision causes the photon wavelength to increase by somewhere between 0 (for a scattering angle of 0°) and twice the Compton wavelength (for a scattering angle of 180°).^[32]

Thomson scattering is the classical elastic quantitative interpretation of the scattering process,^[26] and this can be seen to happen with lower, mid-energy, photons. The classical theory of an electromagnetic wave scattered by charged particles, cannot explain low intensity shifts in wavelength.

Inverse Compton scattering takes place when the electron is moving, and has sufficient kinetic energy compared to the photon. In this case net energy may be transferred from the electron to the photon. The inverse Compton effect is seen in astrophysics when a low energy photon (e.g. of the cosmic microwave background) bounces off a high energy (relativistic) electron. Such electrons are produced in supernovae and active galactic nuclei.^[26]

Møller scattering

Møller scattering Feynman diagram

Mott scattering

Bhabha scattering

Bremsstrahlung scattering

Deep inelastic scattering

Synchrotron emission

If a charged particle such as an electron is accelerated – this can be acceleration in a straight line or motion in a curved path – electromagnetic radiation is emitted by the particle. Within electron storage rings and circular particle accelerators known as synchrotrons, electrons are bent in a circular path and emit X-rays typically. This radially emitted (${\displaystyle \mathbf {a} \perp \mathbf {v} }$) electromagnetic radiation when charged particles are accelerated is called synchrotron radiation.^[33] It is produced in synchrotrons using bending magnets, undulators and/or wigglers.

The first observation came at the General Electric Research Laboratory in Schenectady, New York, on April 24, 1947, in the synchrotron built by a team of Herb Pollack to test the idea of phase-stability principle for RF accelerators.^{[note 8]} When the technician was asked to look around the shielding with a large mirror to check for sparking in the tube, he saw a bright arc of light coming from the electron beam. Robert Langmuir is credited as recognizing it as synchrotron radiation or, as he called it, "Schwinger radiation" after Julian Schwinger.^[34]

Classically, the radiated power P from an accelerated electron is:

${\displaystyle P={\frac {2Ke^{2}}{3c^{2}}}a^{2}}$

this comes from the Larmor formula; where K is an electric permittivity constant,^{[note 9]} e is electron charge, c is the speed of light, and a is the acceleration. Within a circular orbit such as a storage ring, the non-relativistic case is simply the centripetal acceleration. However within a storage ring the acceleration is highly relativistic, and can be obtained as follows:

${\displaystyle a_{non-relativistic}={\frac {v^{2}}{r}}\rightarrow a_{relativistic}={\frac {1}{m}}{\frac {dp}{d\tau }}={\frac {1}{m}}\gamma {\frac {d(\gamma mv)}{dt}}=\gamma ^{2}{\frac {dv}{dt}}=\gamma ^{2}{\frac {v^{2}}{r}}}$,

where v is the circular velocity, r is the radius of the circular accelerator, m is the rest mass of the charged particle, p is the momentum, τ is the Proper time (t/γ), and γ is the Lorentz factor. Radiated power then becomes:

${\displaystyle P={\frac {2Ke^{2}}{3c^{2}}}({\frac {\gamma ^{2}v^{2}}{r}})^{2}={\frac {2Ke^{2}}{3c^{2}}}{\frac {\gamma ^{4}v^{4}}{r^{2}}}}$

For highly relativistic particles, such that velocity becomes nearly constant, the γ⁴ term becomes the dominant variable in determining loss rate, which means that the loss scales as the fourth power of the particle energy γmc²; and the inverse dependence of synchrotron radiation loss on radius argues for building the accelerator as large as possible.^[33]

SLAC

Aerial photo of the Stanford Linear Accelerator Center, with detector complex at the right (east) side

Stanford Linear Accelerator Center is located near Stanford University, California.^[35] Construction began on the 2 mile long linear accelerator in 1962 and was completed in 1967, and in 1968 the first experimental evidence of quarks was discovered resulting in the 1990 Nobel Prize in Physics, shared by SLAC's Richard Taylor and Jerome I. Friedman and Henry Kendall of MIT.^[36] The accelerator came with a 20GeV capacity for the electron acceleration, and while similar to Rutherford's scattering experiment, that experiment operated with alpha particles at only 7MeV. In the SLAC case the incident particle was an electron and the target a proton, and due to the short wavelength of the electron (due to its high energy and momentum) it was able to probe into the proton.^[35] The Stanford Positron Electron Asymmetric Ring (SPEAR) addition to the SLAC made further such discoveries possible, leading to the discovery in 1974 of the J/psi particle, which consists of a paired charm quark and anti-charm quark, and another Nobel Prize in Physics in 1976. This was followed up with Martin Perl's announcement of the discovery of the tau lepton, for which he shared the 1995 Nobel Prize in Physics.^[36]

The SLAC aims to be a premier accelerator laboratory,^[37] to pursue strategic programs in particle physics, particle astrophysics and cosmology, as well as the applications in discovering new drugs for healing, new materials for electronics and new ways to produce clean energy and clean up the environment.^[38] Under the directorship of Chi-Chang Kao the SLAC's fifth director (as of November 2012), a noted X-ray scientist who came to SLAC in 2010 to serve as associate laboratory director for the Stanford Synchrotron Radiation Lightsource.^[39]

RIKEN RI Beam Factory

RIKEN was founded in 1917 as a private research foundation in Tokyo, and is Japan's largest comprehensive research institution. Having grown rapidly in size and scope, it is today renowned for high-quality research in a diverse range of scientific disciplines, and encompasses a network of world-class research centers and institutes across Japan.^[41]

The RIKEN RI Beam Factory, otherwise known as the RIKEN Nishina Centre (for Accelerator-Based Science), is a cyclotron-based research facility which began operating in 2007; 70 years after the first in Japanese cyclotron, from Dr. Yoshio Nishina whose name is given to the facility.^[42]

As of 2006, the facility has a world-class heavy-ion accelerator complex. This consists of a K540-MeV ring cyclotron (RRC) and two different injectors: a variable-frequency heavy-ion linac (RILAC) and a K70-MeV AVF cyclotron (AVF). It has a projectile-fragment separator (RIPS) which provides RI (Radioactive Isotope) beams of less than 60 amu, the world's most intense light-atomic-mass RI beams.^[43]

Overseen by the Nishina Centre, the RI Beam Factory is utilized by users worldwide promoting research in nuclear, particle and hadron physics. This promotion of accelerator applications research is an important mission of the Nishina Centre, and implements the use of both domestic and oversea accelerator facilities.^[44]