Equation of state

From Wikipedia, the free encyclopedia

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

In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most modern equations of state are formulated in the Helmholtz free energy. Equations of state are useful in describing the properties of pure substances and mixtures in liquids, gases, and solid states as well as the state of matter in the interior of stars.

Overview

At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions. An example of an equation of state correlates densities of gases and liquids to temperatures and pressures, known as the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. This equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid.

The general form of an equation of state may be written as

$${\displaystyle f(p,V,T)=0}$$

where ${\displaystyle p}$ is the pressure, ${\displaystyle V}$ the volume, and ${\displaystyle T}$ the temperature of the system. Yet also other variables may be used in that form. It is directly related to Gibbs phase rule, that is, the number of independent variables depends on the number of substances and phases in the system.

An equation used to model this relationship is called an equation of state. In most cases this model will comprise some empirical parameters that are usually adjusted to measurement data. Equations of state can also describe solids, including the transition of solids from one crystalline state to another. Equations of state are also used for the modeling of the state of matter in the interior of stars, including neutron stars, dense matter (quark–gluon plasmas) and radiation fields. A related concept is the perfect fluid equation of state used in cosmology.

Equations of state are applied in many fields such as process engineering and petroleum industry as well as pharmaceutical industry.

Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to the use of the Kelvin (K), with zero being absolute zero.

• ${\displaystyle n}$, number of moles of a substance
• ${\displaystyle V_m}$, ${\displaystyle \frac{V}{n}}$, molar volume, the volume of 1 mole of gas or liquid
• ${\displaystyle R}$, ideal gas constant ≈ 8.3144621 J/mol·K
• ${\displaystyle p_c}$, pressure at the critical point
• ${\displaystyle V_c}$, molar volume at the critical point
• ${\displaystyle T_c}$, absolute temperature at the critical point

Historical background

Boyle's law was one of the earliest formulation of an equation of state. In 1662, the Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:

$${\displaystyle pV=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.}$$

The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.

In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval. This is known today as Charles's law. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:

$${\displaystyle \frac{V_1}{T_1}=\frac{V_2}{T_2}.}$$

Dalton's law (1801) of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.

Mathematically, this can be represented for ${\displaystyle n}$ species as:

$${\displaystyle p_{\text{total}}=p_1+p_2+\cdots +p_n=\sum _{i=1}^n{p}_i.}$$

In 1834, Émile Clapeyron combined Boyle's law and Charles' law into the first statement of the ideal gas law. Initially, the law was formulated as pV_m = R(T_C + 267) (with temperature expressed in degrees Celsius), where R is the gas constant. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with ${\displaystyle 0{}^{\circ }\mathrm{C}=273.15\mathrm{K}}$, giving:

$${\displaystyle p{V}_m=R\left(T_C+273.15{}^{\circ }\text{C}\right).}$$

In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules. His new formula revolutionized the study of equations of state, and was the starting point of cubic equations of state, which most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich-Kwong.

The van der Waals equation of state can be written as

${\displaystyle \left(P+a\frac{1}{V_m^2}\right)(V_m-b)=RT}$

where ${\displaystyle a}$ is a parameter describing the attractive energy between particles and ${\displaystyle b}$ is a parameter describing the volume of the particles.

Ideal gas law

Classical ideal gas law

The classical ideal gas law may be written

$${\displaystyle pV=nRT.}$$

In the form shown above, the equation of state is thus

$${\displaystyle f(p,V,T)=pV-nRT=0.}$$

If the calorically perfect gas approximation is used, then the ideal gas law may also be expressed as follows

$${\displaystyle p=\rho (\gamma -1)e}$$

where ${\displaystyle \rho }$ is the density, ${\displaystyle \gamma =C_p/C_v}$ is the (constant) adiabatic index (ratio of specific heats), ${\displaystyle e=C_{v}T}$ is the internal energy per unit mass (the "specific internal energy"), ${\displaystyle C_v}$ is the specific heat capacity at constant volume, and ${\displaystyle C_p}$ is the specific heat capacity at constant pressure.

Quantum ideal gas law

Since for atomic and molecular gases, the classical ideal gas law is well suited in most cases, let us describe the equation of state for elementary particles with mass ${\displaystyle m}$ and spin ${\displaystyle s}$ that takes into account quantum effects. In the following, the upper sign will always correspond to Fermi–Dirac statistics and the lower sign to Bose–Einstein statistics. The equation of state of such gases with ${\displaystyle N}$ particles occupying a volume ${\displaystyle V}$ with temperature ${\displaystyle T}$ and pressure ${\displaystyle p}$ is given by

$${\displaystyle p=\frac{(2s+1)\sqrt{2m^3{k}_{\text{B}}^5{T}^5}}{3{\pi }^2{\hslash }^3}{\int }_0^{\mathrm{\infty }}\frac{z^{3/2}\mathrm{d}z}{e^{z-\mu /(k_{\text{B}}T)}\pm 1}}$$

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant and ${\displaystyle \mu (T,N/V)}$ the chemical potential is given by the following implicit function

$${\displaystyle \frac{N}{V}=\frac{(2s+1)(m{k}_{\text{B}}T{)}^{3/2}}{\sqrt{2}{\pi }^2{\hslash }^3}{\int }_0^{\mathrm{\infty }}\frac{z^{1/2}\mathrm{d}z}{e^{z-\mu /(k_{\text{B}}T)}\pm 1}.}$$

In the limiting case where ${\displaystyle e^{\mu /(k_{\text{B}}T)}\ll 1}$, this equation of state will reduce to that of the classical ideal gas. It can be shown that the above equation of state in the limit ${\displaystyle e^{\mu /(k_{\text{B}}T)}\ll 1}$ reduces to

$${\displaystyle pV=N{k}_{\text{B}}T\left[1\pm \frac{{\pi }^{3/2}}{2(2s+1)}\frac{N{\hslash }^3}{V(m{k}_{\text{B}}T{)}^{3/2}}+\cdots \right]}$$

With a fixed number density ${\displaystyle N/V}$, decreasing the temperature causes in Fermi gas, an increase in the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in Bose gas, a decrease in pressure from its classical value implying an effective attraction. The quantum nature of this equation is in it dependence on s and ħ.

Cubic equations of state

Main article: Cubic equations of state

Cubic equations of state are called such because they can be rewritten as a cubic function of ${\displaystyle V_m}$. Cubic equations of state originated from the van der Waals equation of state. Hence, all cubic equations of state can be considered 'modified van der Waals equation of state'. There is a very large number of such cubic equations of state. For process engineering, cubic equations of state are today still highly relevant, e.g. the Peng Robinson equation of state or the Soave Redlich Kwong equation of state.

Virial equations of state

Virial equation of state

Main article: Virial expansion

$${\displaystyle \frac{p{V}_m}{RT}=A+\frac{B}{V_m}+\frac{C}{V_m^2}+\frac{D}{V_m^3}+\cdots }$$

Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from statistical mechanics. This equation is also called the Kamerlingh Onnes equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients. A is the first virial coefficient, which has a constant value of 1 and makes the statement that when volume is large, all fluids behave like ideal gases. The second virial coefficient B corresponds to interactions between pairs of molecules, C to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients B, C, D, etc. are functions of temperature only.

The BWR equation of state

Main article: Benedict–Webb–Rubin equation

$${\displaystyle p=\rho RT+\left(B_{0}RT-A_0-\frac{C_0}{T^2}+\frac{D_0}{T^3}-\frac{E_0}{T^4}\right){\rho }^2+\left(bRT-a-\frac{d}{T}\right){\rho }^3+\alpha \left(a+\frac{d}{T}\right){\rho }^6+\frac{c{\rho }^3}{T^2}\left(1+\gamma {\rho }^2\right)\exp \left(-\gamma {\rho }^2\right)}$$

where

• ${\displaystyle p}$ is pressure
• ${\displaystyle \rho }$ is molar density

Values of the various parameters can be found in reference materials. The BWR equation of state has also frequently been used for the modelling of the Lennard-Jones fluid. There are several extensions and modifications of the classical BWR equation of state available.

The Benedict–Webb–Rubin–Starling equation of state is a modified BWR equation of state and can be written as

$${\displaystyle p=\rho RT+\left(B_{0}RT-A_0-\frac{C_0}{T^2}+\frac{D_0}{T^3}-\frac{E_0}{T^4}\right){\rho }^2+\left(bRT-a-\frac{d}{T}+\frac{c}{T^2}\right){\rho }^3+\alpha \left(a+\frac{d}{T}\right){\rho }^6}$$

Note that in this virial equation, the fourth and fifth virial terms are zero. The second virial coefficient is monotonically decreasing as temperature is lowered. The third virial coefficient is monotonically increasing as temperature is lowered.

The Lee–Kesler equation of state is based on the corresponding states principle, and is a modification of the BWR equation of state.

$${\displaystyle p=\frac{RT}{V}\left(1+\frac{B}{V_r}+\frac{C}{V_r^2}+\frac{D}{V_r^5}+\frac{c_4}{T_r^3{V}_r^2}\left(\beta +\frac{\gamma }{V_r^2}\right)\exp \left(\frac{-\gamma }{V_r^2}\right)\right)}$$

Physically based equations of state

There is a large number of physically based equations of state available today. Most of those are formulated in the Helmholtz free energy as a function of temperature, density (and for mixtures additionally the composition). The Helmholtz energy is formulated as a sum of multiple terms modelling different types of molecular interaction or molecular structures, e.g. the formation of chains or dipolar interactions. Hence, physically based equations of state model the effect of molecular size, attraction and shape as well as hydrogen bonding and polar interactions of fluids. In general, physically based equations of state give more accurate results than traditional cubic equations of state, especially for systems containing liquids or solids. Most physically based equations of state are built on monomer term describing the Lennard-Jones fluid or the Mie fluid.

Perturbation theory-based models

Perturbation theory is frequently used for modelling dispersive interactions in an equation of state. There is a large number of perturbation theory based equations of state available today, e.g. for the classical Lennard-Jones fluid. The two most important theories used for these types of equations of state are the Barker-Henderson perturbation theory and the Weeks–Chandler–Andersen perturbation theory.

Statistical associating fluid theory (SAFT)

An important contribution for physically based equations of state is the statistical associating fluid theory (SAFT) that contributes the Helmholtz energy that describes the association (a.k.a. hydrogen bonding) in fluids, which can also be applied for modelling chain formation (in the limit of infinite association strength). The SAFT equation of state was developed using statistical mechanical methods (in particular the perturbation theory of Wertheim) to describe the interactions between molecules in a system. The idea of a SAFT equation of state was first proposed by Chapman et al. in 1988 and 1989. Many different versions of the SAFT models have been proposed, but all use the same chain and association terms derived by Chapman et al.

Multiparameter equations of state

Multiparameter equations of state are empirical equations of state that can be used to represent pure fluids with high accuracy. Multiparameter equations of state are empirical correlations of experimental data and are usually formulated in the Helmholtz free energy. The functional form of these models is in most parts not physically motivated. They can be usually applied in both liquid and gaseous states. Empirical multiparameter equations of state represent the Helmholtz energy of the fluid as the sum of ideal gas and residual terms. Both terms are explicit in temperature and density:

$${\displaystyle \frac{a(T,\rho )}{RT}=\frac{a^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{a}\mathrm{s}}(\tau ,\delta )+a^{\text{residual}}(\tau ,\delta )}{RT}}$$

with

$${\displaystyle \tau =\frac{T_r}{T},\delta =\frac{\rho }{{\rho }_r}}$$

The reduced density ${\displaystyle {\rho }_r}$ and reduced temperature ${\displaystyle T_r}$ are in most cases the critical values for the pure fluid. Because integration of the multiparameter equations of state is not required and thermodynamic properties can be determined using classical thermodynamic relations, there are few restrictions as to the functional form of the ideal or residual terms. Typical multiparameter equations of state use upwards of 50 fluid specific parameters, but are able to represent the fluid's properties with high accuracy. Multiparameter equations of state are available currently for about 50 of the most common industrial fluids including refrigerants. The IAPWS95 reference equation of state for water is also a multiparameter equations of state. Mixture models for multiparameter equations of state exist, as well. Yet, multiparameter equations of state applied to mixtures are known to exhibit artifacts at times.

One example of such an equation of state is the form proposed by Span and Wagner.

$${\displaystyle a^{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}}=\sum _{i=1}^8\sum _{j=-8}^{12}{n}_{i,j}{\delta }^i{\tau }^{j/8}+\sum _{i=1}^5\sum _{j=-8}^{24}{n}_{i,j}{\delta }^i{\tau }^{j/8}\exp \left(-\delta \right)+\sum _{i=1}^5\sum _{j=16}^{56}{n}_{i,j}{\delta }^i{\tau }^{j/8}\exp \left(-{\delta }^2\right)+\sum _{i=2}^4\sum _{j=24}^{38}{n}_{i,j}{\delta }^i{\tau }^{j/2}\exp \left(-{\delta }^3\right)}$$

This is a somewhat simpler form that is intended to be used more in technical applications. Equations of state that require a higher accuracy use a more complicated form with more terms.

List of further equations of state

Stiffened equation of state

When considering water under very high pressures, in situations such as underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence, the stiffened equation of state is often used:

$${\displaystyle p=\rho (\gamma -1)e-\gamma p^0}$$

where ${\displaystyle e}$ is the internal energy per unit mass, ${\displaystyle \gamma }$ is an empirically determined constant typically taken to be about 6.1, and ${\displaystyle p^0}$ is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 2 gigapascals (20,000 atmospheres).

The equation is stated in this form because the speed of sound in water is given by ${\displaystyle c^2=\gamma \left(p+p^0\right)/\rho }$.

Thus water behaves as though it is an ideal gas that is already under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa).

This equation mispredicts the specific heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.

Morse oscillator equation of state

An equation of state of Morse oscillator has been derived, and it has the following form:

${\displaystyle p={\mathrm{\Gamma }}_1\nu +{\mathrm{\Gamma }}_2{\nu }^2}$

Where ${\displaystyle {\mathrm{\Gamma }}_1}$ is the first order virial parameter and it depends on the temperature, ${\displaystyle {\mathrm{\Gamma }}_2}$ is the second order virial parameter of Morse oscillator and it depends on the parameters of Morse oscillator in addition to the absolute temperature. ${\displaystyle \nu }$ is the fractional volume of the system.

Ultrarelativistic equation of state

An ultrarelativistic fluid has equation of state

$${\displaystyle p={\rho }_m{c}_s^2}$$

where ${\displaystyle p}$ is the pressure, ${\displaystyle {\rho }_m}$ is the mass density, and ${\displaystyle c_s}$ is the speed of sound.

Ideal Bose equation of state

The equation of state for an ideal Bose gas is

$${\displaystyle p{V}_m=RT\frac{{\mathrm{Li}}_{\alpha +1}(z)}{\zeta (\alpha )}{\left(\frac{T}{T_c}\right)}^{\alpha }}$$

where α is an exponent specific to the system (e.g. in the absence of a potential field, α = 3/2), z is exp(μ/k_BT) where μ is the chemical potential, Li is the polylogarithm, ζ is the Riemann zeta function, and T_c is the critical temperature at which a Bose–Einstein condensate begins to form.

Jones–Wilkins–Lee equation of state for explosives (JWL equation)

The equation of state from Jones–Wilkins–Lee is used to describe the detonation products of explosives.

$${\displaystyle p=A\left(1-\frac{\omega }{R_{1}V}\right)\exp (-R_{1}V)+B\left(1-\frac{\omega }{R_{2}V}\right)\exp \left(-R_{2}V\right)+\frac{\omega e_0}{V}}$$

The ratio ${\displaystyle V={\rho }_e/\rho }$ is defined by using ${\displaystyle {\rho }_e}$, which is the density of the explosive (solid part) and ${\displaystyle \rho }$, which is the density of the detonation products. The parameters ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle R_1}$, ${\displaystyle R_2}$ and ${\displaystyle \omega }$ are given by several references. In addition, the initial density (solid part) ${\displaystyle {\rho }_0}$, speed of detonation ${\displaystyle V_D}$, Chapman–Jouguet pressure ${\displaystyle P_{CJ}}$ and the chemical energy per unit volume of the explosive ${\displaystyle e_0}$ are given in such references. These parameters are obtained by fitting the JWL-EOS to experimental results. Typical parameters for some explosives are listed in the table below.

Material	${\displaystyle {\rho }_e}$ (g/cm3)	${\displaystyle v_D}$ (m/s)	${\displaystyle p_{CJ}}$ (GPa)	${\displaystyle A}$ (GPa)	${\displaystyle B}$ (GPa)	${\displaystyle R_1}$	${\displaystyle R_2}$	${\displaystyle \omega }$	${\displaystyle e_0}$ (GPa)
TNT	1.630	6930	21.0	373.8	3.747	4.15	0.90	0.35	6.00
Composition B	1.717	7980	29.5	524.2	7.678	4.20	1.10	0.35	8.50
PBX 9501	1.844		36.3	852.4	18.02	4.55	1.3	0.38	10.2

Others

• Tait equation for water and other liquids. Several equations are referred to as the Tait equation.
• Murnaghan equation of state
• Birch–Murnaghan equation of state
• Stacey–Brennan–Irvine equation of state
• Modified Rydberg equation of state
• Adapted polynomial equation of state
• Johnson–Holmquist equation of state
• Mie–Grüneisen equation of state
• Anton-Schmidt equation of state

Linear particle accelerator

From Wikipedia, the free encyclopedia

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

The linac within the Australian Synchrotron uses radio waves from a series of RF cavities at the start of the linac to accelerate the electron beam in bunches to energies of 100 MeV.

A linear particle accelerator (often shortened to linac) is a type of particle accelerator that accelerates charged subatomic particles or ions to a high speed by subjecting them to a series of oscillating electric potentials along a linear beamline. The principles for such machines were proposed by Gustav Ising in 1924, while the first machine that worked was constructed by Rolf Widerøe in 1928 at the RWTH Aachen University. Linacs have many applications: they generate X-rays and high energy electrons for medicinal purposes in radiation therapy, serve as particle injectors for higher-energy accelerators, and are used directly to achieve the highest kinetic energy for light particles (electrons and positrons) for particle physics.

The design of a linac depends on the type of particle that is being accelerated: electrons, protons or ions. Linacs range in size from a cathode ray tube (which is a type of linac) to the 3.2-kilometre-long (2.0 mi) linac at the SLAC National Accelerator Laboratory in Menlo Park, California.

History

Wideroe's linac concept. The voltage from an RF source is connected to a series of tubes which shield the particle between gaps.

Alvarez type linac

In 1924, Gustav Ising published the first description of a linear particle accelerator using a series of accelerating gaps. Particles would proceed down a series of tubes. At a regular frequency, an accelerating voltage would be applied across each gap. As the particles gained speed while the frequency remained constant, the gaps would be spaced farther and farther apart, in order to ensure the particle would see a voltage applied as it reached each gap. Ising never successfully implemented this design.

Rolf Wideroe discovered Ising's paper in 1927, and as part of his PhD thesis, built an 88-inch long, two gap version of the device. Where Ising had proposed a spark gap as the voltage source, Wideroe used a 25kV vacuum tube oscillator. He successfully demonstrated that he had accelerated sodium and potassium ions to an energy of 50 thousand electron volts (50 keV), twice the energy they would have received if accelerated only once by the tube. By successfully accelerating a particle multiple times using the same voltage source, Wideroe demonstrated the utility of radio frequency acceleration.

This type of linac was limited by the voltage sources that were available at the time, and it was not until after World War II that Luis Alvarez was able to use newly developed high frequency oscillators to design the first resonant cavity drift tube linac. An Alvarez linac differs from the Wideroe type in that the RF power is applied to the entire resonant chamber through which the particle travels, and the central tubes are only used to shield the particles during the decelerating portion of the oscillator's phase. Using this approach to acceleration meant that Alvarez's first linac was able to achieve proton energies of 31.5 MeV in 1947, the highest that had ever been reached at the time.

The initial Alvarez type linacs had no strong mechanism for keeping the beam focused, and were limited in length and energy as a result. The development of the strong focusing principle in the early 1950s led to the installation of focusing quadrupole magnets inside the drift tubes, allowing for longer and thus more powerful linacs. Two of the earliest examples of Alvarez linacs with strong focusing magnets were built at CERN and Brookhaven National Laboratory.

In 1947, at about the same time that Alvarez was developing his linac concept for protons, William Hansen constructed the first travelling-wave electron accelerator at Stanford University. Electrons are sufficiently lighter than protons that they achieve speeds close to the speed of light early in the acceleration process. As a result, "accelerating" electrons increase in energy, but can be treated as having a constant velocity from an accelerator design standpoint. This allowed Hansen to use an accelerating structure consisting of a horizontal waveguide loaded by a series of discs. The 1947 accelerator had an energy of 6 MeV. Over time, electron acceleration at the SLAC National Accelerator Laboratory would extend to a size of 2 miles (3.2 km) and an output energy of 50 GeV.

As linear accelerators were developed with higher beam currents, using magnetic fields to focus proton and heavy ion beams presented difficulties for the initial stages of the accelerator. Because the magnetic force is dependent on the particle velocity, it was desirable to create a type of accelerator which could simultaneously accelerate and focus low-to-mid energy hadrons. In 1970, Soviet physicists I. M. Kapchinsky and Vladimir Teplyakov proposed the radio-frequency quadrupole (RFQ) type of accelerating structure. RFQs use vanes or rods with precisely designed shapes in a resonant cavity to produce complex electric fields. These fields provide simultaneous acceleration and focusing to injected particle beams.

Beginning in the 1960s, scientists at Stanford and elsewhere began to explore the use of superconducting radio frequency cavities for particle acceleration. Superconducting cavities made of niobium alloys allowed for much more efficient acceleration, as a substantially higher fraction of the input power could be applied to the beam, rather than lost to heat. Some of the earliest superconducting linacs included the Superconducting Linear Accelerator (for electrons) at Stanford and the Argonne Tandem Linear Accelerator System (for protons and heavy ions) at Argonne National Laboratory.

Basic principles of operation

Animation showing how a linear accelerator works. In this example the particles accelerated (red dots) are assumed to have a positive charge. The graph V(x) shows the electrical potential along the axis of the accelerator at each point in time. The polarity of the RF voltage reverses as the particle passes through each electrode, so when the particle crosses each gap the electric field (E, arrows) has the correct direction to accelerate it. The animation shows a single particle being accelerated each cycle; in actual linacs a large number of particles are injected and accelerated each cycle. The action is shown slowed enormously.

Radiofrequency acceleration

When a charged particle is placed in an electromagnetic field it experiences a force given by the Lorentz force law:

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

(in SI units) where ${\displaystyle q}$ is the charge on the particle, ${\displaystyle \overrightarrow{E}}$ is the electric field, ${\displaystyle \overrightarrow{v}}$ is the particle velocity, and ${\displaystyle \overrightarrow{B}}$ is the magnetic field. The cross product in the magnetic field term means that static magnetic fields cannot be used for particle acceleration, as the magnetic force acts perpendicularly to the direction of particle motion.

As electrostatic breakdown limits the maximum constant voltage which can be applied across a gap to produce an electric field, most accelerators use some form of radiofrequency (RF) acceleration. In RF acceleration, the particle traverses a series of accelerating regions, driven by a source of voltage in such a way that the particle sees an accelerating field as it crosses each region. In this type of acceleration, particles must necessarily travel in "bunches" corresponding to the portion of the oscillator's cycle where the electric field is pointing in the intended direction of acceleration.

If a single oscillating voltage source is used to drive a series of gaps, those gaps must be placed increasingly far apart as the speed of the particle increases. This is to ensure that the particle "sees" the same phase of the oscillator's cycle as it reaches each gap. As particles asymptotically approach the speed of light, the gap separation becomes constant - additional applied force increases the energy of the particles, but does not significantly alter their speed.

Focusing

In order to ensure particles do not escape the accelerator, it is necessary to provide some form of focusing to redirect particles moving away from the central trajectory back towards the intended path. With the discovery of strong focusing, quadrupole magnets are used to actively redirect particles moving away from the reference path. As quadrupole magnets are focusing in one transverse direction and defocusing in the perpendicular direction, it is necessary to use groups of magnets to provide an overall focusing effect in both directions.

Phase stability

Focusing along the direction of travel, also known as phase stability, is an inherent property of RF acceleration. If the particles in a bunch all reach the accelerating region during the rising phase of the oscillating field, then particles which arrive early will see slightly less voltage than the "reference" particle at the center of the bunch. Those particles will therefore receive slightly less acceleration and eventually fall behind the reference particle. Correspondingly, particles which arrive after the reference particle will receive slightly more acceleration, and will catch up to the reference as a result. This automatic correction occurs at each accelerating gap, so the bunch is refocused along the direction of travel each time it is accelerated.

Construction and operation

Quadrupole magnets surrounding the linac of the Australian Synchrotron are used to help focus the electron beam

Building covering the 2 mile (3.2 km) beam tube of the Stanford Linear Accelerator (SLAC) at Menlo Park, California, the second most powerful linac in the world. It has about 80,000 accelerating electrodes and could accelerate electrons to 50 GeV

A linear particle accelerator consists of the following parts:

• A straight hollow pipe vacuum chamber which contains the other components. It is evacuated with a vacuum pump so that the accelerated particles will not collide with air molecules. The length will vary with the application. If the device is used for the production of X-rays for inspection or therapy the pipe may be only 0.5 to 1.5 meters long. If the device is to be an injector for a synchrotron it may be about ten meters long. If the device is used as the primary accelerator for nuclear particle investigations, it may be several thousand meters long.
• The particle source (S) at one end of the chamber which produces the charged particles which the machine accelerates. The design of the source depends on the particle that is being accelerated. Electrons are generated by a cold cathode, a hot cathode, a photocathode, or radio frequency (RF) ion sources. Protons are generated in an ion source, which can have many different designs. If heavier particles are to be accelerated, (e.g., uranium ions), a specialized ion source is needed. The source has its own high voltage supply to inject the particles into the beamline.
• Extending along the pipe from the source is a series of open-ended cylindrical electrodes (C1, C2, C3, C4), whose length increases progressively with the distance from the source. The particles from the source pass through these electrodes. The length of each electrode is determined by the frequency and power of the driving power source and the particle to be accelerated, so that the particle passes through each electrode in exactly one-half cycle of the accelerating voltage. The mass of the particle has a large effect on the length of the cylindrical electrodes; for example an electron is considerably lighter than a proton and so will generally require a much smaller section of cylindrical electrodes as it accelerates very quickly.
• A target (not shown) with which the particles collide, located at the end of the accelerating electrodes. If electrons are accelerated to produce X-rays then a water cooled tungsten target is used. Various target materials are used when protons or other nuclei are accelerated, depending upon the specific investigation. Behind the target are various detectors to detect the particles resulting from the collision of the incoming particles with the atoms of the target. Many linacs serve as the initial accelerator stage for larger particle accelerators such as synchrotrons and storage rings, and in this case after leaving the electrodes the accelerated particles enter the next stage of the accelerator.
• An electronic oscillator and amplifier (G) which generates a radio frequency AC voltage of high potential (usually thousands of volts) which is applied to the cylindrical electrodes. This is the accelerating voltage which produces the electric field which accelerates the particles. As shown, opposite phase voltage is applied to successive electrodes. A high power accelerator will have a separate amplifier to power each electrode, all synchronized to the same frequency.

As shown in the animation, the oscillating voltage applied to alternate cylindrical electrodes has opposite polarity (180° out of phase), so adjacent electrodes have opposite voltages. This creates an oscillating electric field (E) in the gap between each pair of electrodes, which exerts force on the particles when they pass through, imparting energy to them by accelerating them. The particle source injects a group of particles into the first electrode once each cycle of the voltage, when the charge on the electrode is opposite to the charge on the particles. The electrodes are made the correct length so that the accelerating particles take exactly one-half cycle to pass through each electrode. Each time the particle bunch passes through an electrode, the oscillating voltage changes polarity, so when the particles reach the gap between electrodes the electric field is in the correct direction to accelerate them. Therefore, the particles accelerate to a faster speed each time they pass between electrodes; there is little electric field inside the electrodes so the particles travel at a constant speed within each electrode.

The particles are injected at the right time so that the oscillating voltage differential between electrodes is maximum as the particles cross each gap. If the peak voltage applied between the electrodes is ${\displaystyle V_p}$ volts, and the charge on each particle is ${\displaystyle q}$ elementary charges, the particle gains an equal increment of energy of ${\displaystyle q{V}_p}$ electron volts when passing through each gap. Thus the output energy of the particles is

${\displaystyle E=qN{V}_p}$

electron volts, where ${\displaystyle N}$ is the number of accelerating electrodes in the machine.

At speeds near the speed of light, the incremental velocity increase will be small, with the energy appearing as an increase in the mass of the particles. In portions of the accelerator where this occurs, the tubular electrode lengths will be almost constant. Additional magnetic or electrostatic lens elements may be included to ensure that the beam remains in the center of the pipe and its electrodes. Very long accelerators may maintain a precise alignment of their components through the use of servo systems guided by a laser beam.

Concepts in development

Various new concepts are in development as of 2021. The primary goal is to make linear accelerators cheaper, with better focused beams, higher energy or higher beam current.

Induction linear accelerator

Induction linear accelerators use the electric field induced by a time-varying magnetic field for acceleration - like the betatron. The particle beam passes through a series of ring-shaped ferrite cores standing one behind the other, which are magnetized by high-current pulses and in turn each generate an electrical field strength pulse along the axis of the beam direction. Induction linear accelerators are considered for short high current pulses from electrons, but also from heavy ions. The concept goes back to the work of Nicholas Christofilos. Its realization is highly dependent on progress in the development of more suitable Ferrite materials. With electrons, pulse currents of up to 5 kiloamps at energies up to 5 MeV and pulse durations in the range of 20 to 300 nanoseconds were achieved.

Energy Recovery LINAC

In previous electron linear accelerators, the accelerated particles are used only once and then fed into an absorber (beam dump), in which their residual energy is converted into heat. In an Energy Recovery Linac (ERL; literally: "Energy recovery linear accelerator"), instead, the accelerated in resonators and, for example, in undulators. The electrons used are fed back through the accelerator, out of phase by 180 degrees. They therefore pass through the resonators in the decelerating phase and thus return their remaining energy to the field. The concept is comparable to the hybrid drive of motor vehicles, where the kinetic energy released during braking is made available for the next acceleration by charging a battery.

The Brookhaven National Laboratory  and the Helmholtz-Zentrum Berlin with the project "bERLinPro"  reported on corresponding development work . The Berlin experimental accelerator uses superconducting niobium cavity resonators of the type mentioned above. In 2014, three free-electron lasers based on Energy Recovery Linacs were in operation worldwide : in the Jefferson Lab (USA), in the Budker Institute of Nuclear Physics (Russia) and at JAEA (Japan).  At the University of Mainz , an ERL called MESA is under construction and should (as of 2019) go into operation in 2022.

Compact Linear Collider

The concept of the Compact Linear Collider (CLIC) (original name CERN Linear Collider , with the same abbreviation) for electrons and positrons provides a traveling wave accelerator for energies of the order of 1 tera-electron volt (TeV). Instead of the otherwise necessary numerous klystron amplifiers to generate the acceleration power, a second, parallel electron linear accelerator of lower energy is to be used, which works with superconducting cavities in which standing waves are formed. High-frequency power is extracted from it at regular intervals and transmitted to the main accelerator. In this way, the very high acceleration field strength of 80 MV / m should be achieved.

Kielfeld accelerator (plasma accelerator)

In cavity resonators, the dielectric strength limits the maximum acceleration that can be achieved within a certain distance. This limit can be circumvented using accelerated waves in plasma to generate the accelerating field in Kielfeld accelerators: A laser or particle beam excites an oscillation in a plasma, which is associated with very strong electric field strengths. This means that significantly (factors of 100s to 1000s ) more compact linear accelerators can possibly be built. Experiments involving high power lasers in metal vapour plasmas suggest that a beam line length reduction from some tens of metres to a few cm is quite possible.

Compact Medical Accelerators

The LIGHT program (Linac for Image-Guided Hadron Therapy) hopes to create a design capable of accelerating protons to 200MeV or so for medical use over a distance of a few tens of metres, by optimising and nesting existing accelerator techniques. The current design (2020) uses the highest practical bunch frequency (currently ~ 3 GHz) for a Radio-frequency quadrupole (RFQ) stage from injection at 50kVdC to ~5MeV bunches, a Side Coupled Drift Tube Linac (SCDTL) to accelerate from 5Mev to ~ 40MeV and a Cell Coupled Linac (CCL) stage final, taking the output to 200-230MeV. Each stage is optimised to allow close coupling and synchronous operation during the beam energy build-up. The project aim is to make proton therapy a more accessible mainstream medicine as an alternative to existing radio therapy.

Modern linear accelerator concepts

The higher the frequency of the acceleration voltage selected, the more individual acceleration thrusts per path length a particle of a given speed experiences, and the shorter the accelerator can therefore be overall. That is why accelerator technology developed in the pursuit of higher particle energies, especially towards higher frequencies.

The linear accelerator concepts (often called accelerator structures in technical terms) that have been used since around 1950 work with frequencies in the range from around 100 MHz to a few gigahertz (GHz) and use the electric field component of electromagnetic waves.

Standing waves and traveling waves

When it comes to energies of more than a few MeV, accelerators for ions are different from those for electrons. The reason for this is the large mass difference between the particles. Electrons are already close to the speed of light, the absolute speed limit, at a few MeV; with further acceleration, as described by relativistic mechanics, almost only their energy and momentum increase. On the other hand, with ions of this energy range, the speed also increases significantly due to further acceleration.

The acceleration concepts used today for ions are always based on electromagnetic standing waves that are formed in suitable resonators. Depending on the type of particle, energy range and other parameters, very different types of resonators are used; the following sections only cover some of them. Electrons can also be accelerated with standing waves above a few MeV. An advantageous alternative here, however, is a progressive wave, a traveling wave. The phase velocity the traveling wave must be roughly equal to the particle speed. Therefore, this technique is only suitable when the particles are almost at the speed of light, so that their speed only increases very little.

The development of high-frequency oscillators and power amplifiers from the 1940s, especially the klystron, was essential for these two acceleration techniques . The first larger linear accelerator with standing waves - for protons - was built in 1945/46 in the Lawrence Berkeley National Laboratory under the direction of Luis W. Alvarez. The frequency used was 200 MHz.  The first electron accelerator with traveling waves of around 2 GHz was developed a little later at Stanford University by W.W. Hansen and colleagues.

Principle of the acceleration of particle packets

by a standing wave

by a traveling wave

In the two diagrams, the curve and arrows indicate the force acting on the particles. Only at the points with the correct direction of the electric field vector, i.e. the correct direction of force, can particles absorb energy from the wave. (An increase in speed cannot be seen in the scale of these images.)

Advantages

The Stanford University superconducting linear accelerator, housed on campus below the Hansen Labs until 2007. This facility is separate from SLAC

Steel casting undergoing x-ray using the linear accelerator at Goodwin Steel Castings Ltd

The linear accelerator could produce higher particle energies than the previous electrostatic particle accelerators (the Cockcroft-Walton accelerator and Van de Graaff generator) that were in use when it was invented. In these machines, the particles were only accelerated once by the applied voltage, so the particle energy in electron volts was equal to the accelerating voltage on the machine, which was limited to a few million volts by insulation breakdown. In the linac, the particles are accelerated multiple times by the applied voltage, so the particle energy is not limited by the accelerating voltage.

High power linacs are also being developed for production of electrons at relativistic speeds, required since fast electrons traveling in an arc will lose energy through synchrotron radiation; this limits the maximum power that can be imparted to electrons in a synchrotron of given size. Linacs are also capable of prodigious output, producing a nearly continuous stream of particles, whereas a synchrotron will only periodically raise the particles to sufficient energy to merit a "shot" at the target. (The burst can be held or stored in the ring at energy to give the experimental electronics time to work, but the average output current is still limited.) The high density of the output makes the linac particularly attractive for use in loading storage ring facilities with particles in preparation for particle to particle collisions. The high mass output also makes the device practical for the production of antimatter particles, which are generally difficult to obtain, being only a small fraction of a target's collision products. These may then be stored and further used to study matter-antimatter annihilation.

Medical linacs

Historical image showing Gordon Isaacs, the first patient treated for retinoblastoma with linear accelerator radiation therapy (in this case an electron beam), in 1957, in the U.S. Other patients had been treated by linac for other diseases since 1953 in the UK. Gordon's right eye was removed on January 11, 1957 because cancer had spread there. His left eye, however, had only a localized tumor that prompted Henry Kaplan to treat it with the electron beam.

Linac-based radiation therapy for cancer treatment began with the first patient treated in 1953 in London, UK, at the Hammersmith Hospital, with an 8 MV machine built by Metropolitan-Vickers and installed in 1952, as the first dedicated medical linac. A short while later in 1954, a 6 MV linac was installed in Stanford, USA, which began treatments in 1956.

Medical linear accelerators accelerate electrons using a tuned-cavity waveguide, in which the RF power creates a standing wave. Some linacs have short, vertically mounted waveguides, while higher energy machines tend to have a horizontal, longer waveguide and a bending magnet to turn the beam vertically towards the patient. Medical linacs use monoenergetic electron beams between 4 and 25 MeV, giving an X-ray output with a spectrum of energies up to and including the electron energy when the electrons are directed at a high-density (such as tungsten) target. The electrons or X-rays can be used to treat both benign and malignant disease. The LINAC produces a reliable, flexible and accurate radiation beam. The versatility of LINAC is a potential advantage over cobalt therapy as a treatment tool. In addition, the device can simply be powered off when not in use; there is no source requiring heavy shielding – although the treatment room itself requires considerable shielding of the walls, doors, ceiling etc. to prevent escape of scattered radiation. Prolonged use of high powered (>18 MeV) machines can induce a significant amount of radiation within the metal parts of the head of the machine after power to the machine has been removed (i.e. they become an active source and the necessary precautions must be observed).

Aerial view of the Little LINAC Model

In 2019 a Little Linac model kit, containing 82 building blocks, was developed for children undergoing radiotherapy treatment for cancer. The hope is that building the model will alleviate some of the stress experienced by the child before undergoing treatment by helping them to understand what the treatment entails. The kit was developed by Professor David Brettle, Institute of Physics and Engineering in Medicine (IPEM) in collaboration with manufacturers Best-Lock Ltd. The model can be seen at the Science Museum, London.

Application for medical isotope development

The expected shortages of Mo-99, and the technetium-99m medical isotope obtained from it, have also shed light onto linear accelerator technology to produce Mo-99 from non-enriched Uranium through neutron bombardment. This would enable the medical isotope industry to manufacture this crucial isotope by a sub-critical process. The aging facilities, for example the Chalk River Laboratories in Ontario, Canada, which still now produce most Mo-99 from highly enriched uranium could be replaced by this new process. In this way, the sub-critical loading of soluble uranium salts in heavy water with subsequent photo neutron bombardment and extraction of the target product, Mo-99, will be achieved.

Disadvantages

• The device length limits the locations where one may be placed.
• A great number of driver devices and their associated power supplies are required, increasing the construction and maintenance expense of this portion.
• If the walls of the accelerating cavities are made of normally conducting material and the accelerating fields are large, the wall resistivity converts electric energy into heat quickly. On the other hand, superconductors also need constant cooling to keep them below their critical temperature, and the accelerating fields are limited by quenches. Therefore, high energy accelerators such as SLAC, still the longest in the world (in its various generations), are run in short pulses, limiting the average current output and forcing the experimental detectors to handle data coming in short bursts.