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107-501: The original concept was developed in 1954 by N.V. Filippov, who noticed the effect while working on early pinch machines in the USSR. A major research program on DPF was carried out in the USSR through the late 1950s, and continues to this day. A different version of the same basic concept was independently discovered in the US by J.W. Mather in the early 1960s. This version saw some development in

214-444: A l = γ ⋅ p ρ = γ ⋅ R ⋅ T M = γ ⋅ k ⋅ T m , {\displaystyle c_{\mathrm {ideal} }={\sqrt {\gamma \cdot {p \over \rho }}}={\sqrt {\gamma \cdot R\cdot T \over M}}={\sqrt {\gamma \cdot k\cdot T \over m}},} where This equation applies only when

321-402: A dispersive medium , the speed of sound is a function of sound frequency, through the dispersion relation . Each frequency component propagates at its own speed, called the phase velocity , while the energy of the disturbance propagates at the group velocity . The same phenomenon occurs with light waves; see optical dispersion for a description. The speed of sound is variable and depends on

428-430: A shock wave that would continue the process after the current was removed. In the stabilized pinch , new magnetic fields would be added that would mix with the current's field and create a more stable configuration. In testing, neither of these systems worked, and the pinch route to fusion was largely abandoned by the early 1960s. During experiments on a linear pinch machine, Filippov noticed that certain arrangements of

535-449: A 2007 Google Tech Talk. On November 14, 2008, Lerner received funding for continued research, to test the scientific feasibility of Focus Fusion. On October 15, 2009, the DPF device "Focus Fusion-1" achieved its first pinch. On January 28, 2011, LPP published initial results including experimental shots with considerably higher fusion yields than the historical DPF trend. In March, 2012,

642-413: A certain value if the plasma focus is to operate efficiently. The critical 'speed' design parameter for neutron-producing devices is I a p {\displaystyle {\frac {I}{a{\sqrt {p}}}}} , where I {\displaystyle I} is the current, a {\displaystyle a} is the anode radius, and p {\displaystyle p}

749-436: A compressed and distorted length of copper tube from a lightning rod after it had been struck by lightning. Their analysis showed that the forces due to the interaction of the large current flow with its own magnetic field could have caused the compression and distortion. A similar, and apparently independent, theoretical analysis of the pinch effect in liquid metals was published by Northrup in 1907. The next major development

856-453: A compression wave in a fluid is determined by the medium's compressibility and density . In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor of shear modulus which affects compression waves due to off-axis elastic energies which are able to influence effective tension and relaxation in a compression. The speed of shear waves, which can occur only in solids,

963-410: A computation of the speed of sound in air as 979 feet per second (298 m/s). This is too low by about 15%. The discrepancy is due primarily to neglecting the (then unknown) effect of rapidly fluctuating temperature in a sound wave (in modern terms, sound wave compression and expansion of air is an adiabatic process , not an isothermal process ). This error was later rectified by Laplace . During

1070-473: A deuterium fill pressure of 4 Torr (530 Pa). The length of the anode has then to be matched to the risetime of the capacitor current in order to allow an average axial transit speed of the current sheath of just over 50 mm/μs. Thus a capacitor risetime of 3 μs requires a matched anode length of 160 mm. The above example of peak current of 180 kA rising in 3 μs, anode radius and length of respectively 10 and 160 mm are close to

1177-454: A dynamic Z-pinch. In plasma physics three pinch geometries are commonly studied: the θ-pinch, the Z-pinch , and the screw pinch. These are cylindrically shaped. The cylinder is symmetric in the axial ( z ) direction and the azimuthal (θ) directions. The one-dimensional pinches are named for the direction the current travels. The θ-pinch has a magnetic field directed in the z direction and

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1284-460: A large burst of current is applied to a dilute gas inside the tube. This current initially ionizes the gas into a plasma. Once the ionization is complete, which occurs in microseconds, the plasma begins to conduct a current. Due to the Lorentz force , this current creates a magnetic field that causes the plasma to "pinch" itself down into a filament, similar to a lightning bolt. This process increases

1391-476: A large diamagnetic current directed in the θ direction. Using Ampère's circuital law (discarding the displacement term) Since B is only a function of r we can simplify this to So J points in the θ direction. Thus, the equilibrium condition ( ∇ p = j × B {\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} } ) for the θ-pinch reads: θ-pinches tend to be resistant to plasma instabilities; This

1498-486: A pipe aligned with the x {\displaystyle x} axis and with a cross-sectional area of A {\displaystyle A} . In time interval d t {\displaystyle dt} it moves length d x = v d t {\displaystyle dx=v\,dt} . In steady state , the mass flow rate m ˙ = ρ v A {\displaystyle {\dot {m}}=\rho vA} must be

1605-407: A reflected shock front emanates from the axis until it meets the driving current sheath which then forms the axisymmetric boundary of the pinched, or focused, hot plasma column. The dense plasma column (akin to the Z-pinch ) rapidly pinches and undergoes instabilities and breaks up. The intense electromagnetic radiation and particle bursts, collectively referred to as multi-radiation occur during

1712-459: A single given gas (assuming the molecular weight does not change) and over a small temperature range (for which the heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas. In non-ideal gas behavior regimen, for which the Van der Waals gas equation would be used, the proportionality is not exact, and there is a slight dependence of sound velocity on

1819-405: A small table-top-sized plasma focus machine produces essentially the same plasma characteristics (temperature and density) as the largest plasma focus. Of course the larger machine will produce the larger volume of focused plasma with a corresponding longer lifetime and more radiation yield. Even the smallest plasma focus has essentially the same dynamic characteristics as larger machines, producing

1926-472: A steady state with forces balancing: where ∇ p is the magnetic pressure gradient, and p e and p i are the electron and ion pressures, respectively. Then using Maxwell's equation ∇ × B = μ 0 j {\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} } and the ideal gas law p = N k T {\displaystyle p=NkT} , we derive: where N

2033-401: A theta pinch able to crush an aluminium soft drink can using the Lorentz forces created when large currents are induced in the can by the strong magnetic field of the primary coil. An electromagnetic aluminium can crusher consists of four main components: a high-voltage DC power supply , which provides a source of electrical energy , a large energy discharge capacitor to accumulate

2140-404: A vertical post extending off the end of the inner electrode. In this area the density is greatly increased. The whole process proceeds at many times the speed of sound in the ambient gas. As the current sheath continues to move axially, the portion in contact with the anode slides across the face of the anode, axisymmetrically. When the imploding front of the shock wave coalesces onto the axis,

2247-687: A very short axial phase compared to a Mather focus. When operated using deuterium , intense bursts of X-rays and charged particles are emitted, as are nuclear fusion byproducts including neutrons . There is ongoing research that demonstrates potential applications as a soft X-ray source for next-generation microelectronics lithography , surface micromachining , pulsed X-ray and neutron source for medical and security inspection applications and materials modification, among others. For nuclear weapons applications, dense plasma focus devices can be used as an external neutron source . Other applications include simulation of nuclear explosions (for testing of

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2354-419: Is associated with compression and decompression in the direction of travel, and is the same process in gases and liquids, with an analogous compression-type wave in solids. Only compression waves are supported in gases and liquids. An additional type of wave, the transverse wave , also called a shear wave , occurs only in solids because only solids support elastic deformations. It is due to elastic deformation of

2461-417: Is called the object's Mach number . Objects moving at speeds greater than the speed of sound ( Mach 1 ) are said to be traveling at supersonic speeds . In Earth's atmosphere, the speed of sound varies greatly from about 295 m/s (1,060 km/h; 660 mph) at high altitudes to about 355 m/s (1,280 km/h; 790 mph) at high temperatures. Sir Isaac Newton 's 1687 Principia includes

2568-412: Is determined by the medium's compressibility , shear modulus , and density. The speed of shear waves is determined only by the solid material's shear modulus and density. In fluid dynamics , the speed of sound in a fluid medium (gas or liquid) is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound (in the same medium)

2675-811: Is determined simply by the solid material's shear modulus and density. The speed of sound in mathematical notation is conventionally represented by c , from the Latin celeritas meaning "swiftness". For fluids in general, the speed of sound c is given by the Newton–Laplace equation: c = K s ρ , {\displaystyle c={\sqrt {\frac {K_{s}}{\rho }}},} where K s = ρ ( ∂ P ∂ ρ ) s {\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}} , where P {\displaystyle P}

2782-497: Is due in part to Alfvén's theorem (also known as the frozen-in flux theorem). The Z-pinch has a magnetic field in the θ direction and a current J flowing in the z direction. Again, by electrostatic Ampère's law, Thus, the equilibrium condition, ∇ p = j × B {\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} } , for the Z-pinch reads: Although Z-pinches satisfy

2889-577: Is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode have energies that are too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See

2996-533: Is given here again via Ampère's law: A common problem with one-dimensional pinches is the end losses. Most of the motion of particles is along the magnetic field. With the θ-pinch and the screw-pinch, this leads particles out of the end of the machine very quickly, leading to a loss of mass and energy. Along with this problem, the Z-pinch has major stability problems. Though particles can be reflected to some extent with magnetic mirrors , even these allow many particles to pass. A common method of beating these end losses,

3103-452: Is in the order of 5 ⋅ 10 10 {\displaystyle {5\cdot 10^{10}}} J/m. For example, for a capacitor bank of 3kJ, the anode radius is in the order of 12mm. This parameter has a range of 3.6x10^9 to 7.6x10^11 for the machines surveyed by Soto. The wide range of this parameter is because it is a "storage energy density" which translates into plasma energy density with different efficiency depending on

3210-669: Is one with energy capacity of 1 MJ making it one of the largest plasma focus devices in the world. In Argentina there is an Inter-institutional Program for Plasma Focus Research since 1996, coordinated by a National Laboratory of Dense Magnetized Plasmas ( www.pladema.net ) in Tandil, Buenos Aires. The Program also cooperates with the Chilean Nuclear Energy Commission, and networks the Argentine National Energy Commission,

3317-609: Is the compression of an electrically conducting filament by magnetic forces, or a device that does such. The conductor is usually a plasma , but could also be a solid or liquid metal . Pinches were the first type of device used for experiments in controlled nuclear fusion power . Pinches occur naturally in electrical discharges such as lightning bolts , planetary auroras , current sheets , and solar flares . Pinches exist in nature and in laboratories. Pinches differ in their geometry and operating forces. These include: Pinches may become unstable . They radiate energy across

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3424-451: Is the gas density or pressure. For example, for neutron-optimised operation in deuterium the value of this critical parameter, experimentally observed over a range of machines from kilojoules to hundreds of kilojoules, is: 9 kA/(mm·Torr), or 780 kA/(m·Pa), with a remarkably small deviation of 10% over such a large range of sizes of machines. Thus if we have a peak current of 180 kA we require an anode radius of 10 mm with

3531-551: Is the number of electrons per unit length along the axis, T e and T i are the electron and ion temperatures, I is the total beam current, and k is the Boltzmann constant . The generalized Bennett relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω. Along the axis of the plasma cylinder flows a current density j z , resulting in an azimuthal magnetic field Β φ . Originally derived by Witalis,

3638-472: Is the pressure and the derivative is taken isentropically, that is, at constant entropy s . This is because a sound wave travels so fast that its propagation can be approximated as an adiabatic process , meaning that there isn't enough time, during a pressure cycle of the sound, for significant heat conduction and radiation to occur. Thus, the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of

3745-460: Is to bend the cylinder around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar paths on the outer portion (outboard side). Thus, a new theory is needed. This gives rise to the famous Grad–Shafranov equation . Numerical solutions to the Grad–Shafranov equation have also yielded some equilibria, most notably that of

3852-551: The B field has a θ component and a z component So this time J has a component in the z direction and a component in the θ direction. Finally, the equilibrium condition ( ∇ p = j × B {\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} } ) for the screw pinch reads: The screw pinch might be produced in laser plasma by colliding optical vortices of ultrashort duration. For this purpose optical vortices should be phase-conjugated. The magnetic field distribution

3959-446: The ozone layer . This produces a positive speed of sound gradient in this region. Still another region of positive gradient occurs at very high altitudes, in the thermosphere above 90 km . For an ideal gas, K (the bulk modulus in equations above, equivalent to C , the coefficient of stiffness in solids) is given by K = γ ⋅ p . {\displaystyle K=\gamma \cdot p.} Thus, from

4066-526: The reversed field pinch . As of 2015 , there is no coherent analytical theory for three-dimensional equilibria. The general approach to finding such equilibria is to solve the vacuum ideal MHD equations. Numerical solutions have yielded designs for stellarators . Some machines take advantage of simplification techniques such as helical symmetry (for example University of Wisconsin's Helically Symmetric eXperiment). However, for an arbitrary three-dimensional configuration, an equilibrium relation, similar to that of

4173-548: The springs , and the mass of the spheres. As long as the spacing of the spheres remains constant, stiffer springs/bonds transmit energy more quickly, while more massive spheres transmit energy more slowly. In a real material, the stiffness of the springs is known as the " elastic modulus ", and the mass corresponds to the material density . Sound will travel more slowly in spongy materials and faster in stiffer ones. Effects like dispersion and reflection can also be understood using this model. Some textbooks mistakenly state that

4280-614: The "One o'Clock Gun" is fired at the eastern end of Edinburgh Castle. Standing at the base of the western end of the Castle Rock, the sound of the Gun can be heard through the rock, slightly before it arrives by the air route, partly delayed by the slightly longer route. It is particularly effective if a multi-gun salute such as for "The Queen's Birthday" is being fired. In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. A longitudinal wave

4387-461: The "plasma energy density". On the other hand, another proposed, so called "energy density parameter" 28 E a 3 {\displaystyle {28E \over a^{3}}} , where E is the energy stored in the capacitor bank and a is the anode radius, for neutron-optimised operation in deuterium the value of this critical parameter, experimentally observed over a range of machines from tens of joules to hundreds of kilojoules,

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4494-432: The 1-D configurations exists: Where κ is the curvature vector defined as: with b the unit vector tangent to B . Consider a cylindrical column of fully ionized quasineutral plasma, with an axial electric field, producing an axial current density, j , and associated azimuthal magnetic field, B . As the current flows through its own magnetic field, a pinch is generated with an inward radial force density of j x B . In

4601-590: The 17th century there were several attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630 (1,380 Parisian feet per second), Pierre Gassendi in 1635 (1,473 Parisian feet per second) and Robert Boyle (1,125 Parisian feet per second). In 1709, the Reverend William Derham , Rector of Upminster, published a more accurate measure of the speed of sound, at 1,072 Parisian feet per second. (The Parisian foot

4708-399: The 1970s, and variations continue to be developed. The basic design derives from the z-pinch concept. Both the DPF and pinch use large electrical currents run through a gas to cause it to ionize into a plasma and then pinch down on itself to increase the density and temperature of the plasma. The DPF differs largely in form; most devices use two concentric cylinders and form the pinch at

4815-419: The DPF requires that the bremsstrahlung losses be reduced by quantum mechanical effects induced by an extremely strong magnetic field " frozen into the plasma" . The high magnetic field also results in a high rate of emission of cyclotron radiation , but at the densities envisioned, where the plasma frequency is larger than the cyclotron frequency , most of this power will be reabsorbed before being lost from

4922-470: The MHD equilibrium condition, it is important to note that this is an unstable equilibrium, resulting in various instabilies such as the m = 0 instability ('sausage'), m = 1 instability ('kink'), and various other higher order instabilities. The screw pinch is an effort to combine the stability aspects of the θ-pinch and the confinement aspects of the Z-pinch. Referring once again to Ampère's law, But this time,

5029-750: The NX3 in Singapore, the first plasma focus to be commissioned in a US university in recent times, the KSU Plasma Focus at Kansas State University which recorded its first fusion neutron emitting pinch on New Year's Eve 2009 and the IR-MPF-100 plasma focus (115kJ) in Iran. Several groups proposed that fusion power based on the DPF could be economically viable, possibly even with low-neutron fuel cycles like p-B11. The feasibility of net power from p-B11 in

5136-405: The Newton–Laplace equation above, the speed of sound in an ideal gas is given by c = γ ⋅ p ρ , {\displaystyle c={\sqrt {\gamma \cdot {p \over \rho }}},} where Using the ideal gas law to replace p with nRT / V , and replacing ρ with nM / V , the equation for an ideal gas becomes c i d e

5243-669: The Scientific Council of Buenos Aires, the University of Center, the University of Mar del Plata, The University of Rosario, and the Institute of Plasma Physics of the University of Buenos Aires. The program operates six Plasma Focus Devices, developing applications, in particular ultra-short tomography and substance detection by neutron pulsed interrogation. PLADEMA also contributed during the last decade with several mathematical models of Plasma Focus. The thermodynamic model

5350-407: The axial magnetic field B z . The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside. Chart regions: (a) In the top-left region, the pinching force dominates. (b) Towards the bottom, outward kinetic pressures balance inwards magnetic pressure, and the total pressure is constant. (c) To the right of the vertical line Δ W B z  = 0,

5457-790: The bigger devices (driven by generators of 1MJ). However, the stability of the plasma pinch highly depends on the size and energy of the device. A rich plasma phenomenology it has been observed in the table-top plasma focus devices developed at the Chilean Nuclear Energy Commission: filamentary structures, toroidal singularities, plasma bursts and plasma jets generations. In addition, possible applications are explored using these kind of small plasma devices: development of portable generator as non-radioactive sources of neutrons and X-rays for field applications, pulsed radiation applied to biological studies, plasma focus as neutron source for nuclear fusion-fission hybrid reactors, and

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5564-518: The characters to begin their heist. Speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At 20 °C (68 °F), the speed of sound in air is about 343  m/s (1,125  ft/s ; 1,235  km/h ; 767  mph ; 667  kn ), or 1  km in 2.91 s or one mile in 4.69 s . It depends strongly on temperature as well as

5671-402: The company announced that it had achieved temperatures of 1.8 billion degrees, beating the old record of 1.1 billion that had survived since 1978. In 2016 the company announced that it had achieved a fusion yield of 0.25 joules. In 2017 the company reduced impurities by mass by 3x and ion numbers by 10x. Fusion yield increased by 50%. Fusion yield doubled compared to other plasma focus devices with

5778-458: The current in order to maximize the resulting pinch, and to ensure that the device works safely. For more details, see the notes. The first creation of a Z-pinch in the laboratory may have occurred in 1790 in Holland when Martinus van Marum created an explosion by discharging 100 Leyden jars into a wire. The phenomenon was not understood until 1905, when Pollock and Barraclough investigated

5885-481: The dense plasma and breakup phases. These critical phases last typically tens of nanoseconds for a small (kJ, 100 kA) focus machine to around a microsecond for a large (MJ, several MA) focus machine. The process, including axial and radial phases, may last, for the Mather DPF machine, a few microseconds (for a small focus) to 10 microseconds for a larger focus machine. A Filippov focus machine has

5992-456: The denser materials. An illustrative example of the two effects is that sound travels only 4.3 times faster in water than air, despite enormous differences in compressibility of the two media. The reason is that the greater density of water, which works to slow sound in water relative to the air, nearly makes up for the compressibility differences in the two media. For instance, sound will travel 1.59 times faster in nickel than in bronze, due to

6099-408: The density of the plasma very rapidly, causing its temperature to increase. Early devices quickly demonstrated a problem with the stability of this process. As the current began to flow in the plasma, magnetic effects known as the "sausage" and "kink" appeared that caused the plasma to become unstable and eventually hit the sides of the container. When this occurred, the hot plasma would cause atoms of

6206-461: The design parameters of the UNU/ICTP PFF (United Nations University/International Centre for Theoretical Physics Plasma Fusion Facility). This small table-top device was designed as a low-cost integrated experimental system for training and transfer to initiate/strengthen experimental plasma research in developing countries. It can be noted that the square of the drive parameter is a measure of

6313-505: The earliest systems to be seriously developed for fusion research, starting with very small machines built in London in 1948. These normally took one of two forms; linear pinch machines are straight tubes with electrodes at both ends to apply the current into the plasma, whereas toroidal pinch machines are donut-shaped machines with large magnets wrapped around them that supply the current via magnetic induction . In both types of machines,

6420-426: The electrical energy, a high voltage switch or spark gap , and a robust coil (capable of surviving high magnetic pressure) through which the stored electrical energy can be quickly discharged in order to generate a correspondingly strong pinching magnetic field (see diagram below). In practice, such a device is somewhat more sophisticated than the schematic diagram suggests, including electrical components that control

6527-459: The electrode formed a much thinner 10 nm oxide layer with correspondingly fewer impurities and less electrode erosion than with tungsten electrodes. Fusion yield reached 0.1 joule. Yield generally increased and impurities decreased with an increasing number of shots. Pinch (plasma physics) A pinch (or: Bennett pinch (after Willard Harrison Bennett ), electromagnetic pinch , magnetic pinch , pinch effect , or plasma pinch . )

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6634-404: The electrodes and tube would cause the plasma to form into new shapes. This led to the DPF concept. In a typical DPF machine, there are two cylindrical electrodes. The inner one, often solid, is physically separated from the outer by an insulating disk at one end of the device. It is left open at the other end. The end result is something like a coffee mug with a half hot dog standing on its end in

6741-441: The electronic equipment) and a short and intense neutron source useful for non-contact discovery or inspection of nuclear materials (uranium, plutonium). An important characteristic of the dense plasma focus is that the energy density of the focused plasma is practically a constant over the whole range of machines, from sub-kilojoule machines to megajoule machines, when these machines are tuned for optimal operation. This means that

6848-462: The end of the central cylinder. In contrast, z-pinch systems generally use a single cylinder, sometimes a torus, and pinch the plasma into the center. The plasma focus is similar to the high-intensity plasma gun device (HIPGD) (or just plasma gun ), which ejects plasma in the form of a plasmoid, without pinching it. A comprehensive review of the dense plasma focus and its diverse applications has been made by Krishnan in 2012. Pinch-based devices are

6955-436: The end, it continues moving for a short time, but the endpoints of the current sheet remain attached to the end of the cylinders. This causes the plasma sheet to bow out into a shape not unlike an umbrella or the cap of a mushroom. At this point further movement stops, and the continuing current instead begins to pinch the section near the central electrode. Eventually this causes the former ring-shaped area to compress down into

7062-445: The fastest it can travel under normal conditions. In theory, the speed of sound is actually the speed of vibrations. Sound waves in solids are composed of compression waves (just as in gases and liquids) and a different type of sound wave called a shear wave , which occurs only in solids. Shear waves in solids usually travel at different speeds than compression waves, as exhibited in seismology . The speed of compression waves in solids

7169-569: The form and is applicable to many space plasmas. The Carlqvist relation can be illustrated (see right), showing the total current ( I ) versus the number of particles per unit length ( N ) in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold ( T i = T e = T n = 20 K), containing mainly hydrogen with a mean particle mass 3×10 kg. The thermokinetic energy W k >> πa p k (a). The curves, ΔW Bz show different amounts of excess magnetic energy per unit length due to

7276-477: The gas pressure. Humidity has a small but measurable effect on the speed of sound (causing it to increase by about 0.1%–0.6%), because oxygen and nitrogen molecules of the air are replaced by lighter molecules of water . This is a simple mixing effect. In the Earth's atmosphere , the chief factor affecting the speed of sound is the temperature . For a given ideal gas with constant heat capacity and composition,

7383-413: The generalized Bennett relation results in: The positive terms in the equation are expansional forces while the negative terms represent beam compressional forces. The Carlqvist relation, published by Per Carlqvist in 1988, is a specialization of the generalized Bennett relation (above), for the case that the kinetic pressure is much smaller at the border of the pinch than in the inner parts. It takes

7490-615: The greater stiffness of nickel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen ( protium ) gas than in heavy hydrogen ( deuterium ) gas, since deuterium has similar properties but twice the density. At the same time, "compression-type" sound will travel faster in solids than in liquids, and faster in liquids than in gases, because the solids are more difficult to compress than liquids, while liquids, in turn, are more difficult to compress than gases. A practical example can be observed in Edinburgh when

7597-401: The ground, creating an acoustic shadow at some distance from the source. The decrease of the speed of sound with height is referred to as a negative sound speed gradient . However, there are variations in this trend above 11 km . In particular, in the stratosphere above about 20 km , the speed of sound increases with height, due to an increase in temperature from heating within

7704-413: The gunshot with a half-second pendulum. Measurements were made of gunshots from a number of local landmarks, including North Ockendon church. The distance was known by triangulation , and thus the speed that the sound had travelled was calculated. The transmission of sound can be illustrated by using a model consisting of an array of spherical objects interconnected by springs. In real material terms,

7811-466: The important factors, since fluids do not transmit shear stresses. In heterogeneous fluids, such as a liquid filled with gas bubbles, the density of the liquid and the compressibility of the gas affect the speed of sound in an additive manner, as demonstrated in the hot chocolate effect . In gases, adiabatic compressibility is directly related to pressure through the heat capacity ratio (adiabatic index), while pressure and density are inversely related to

7918-482: The magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve Δ W B z  = 0, the gravitational force is negligible. Note that the chart shows a special case of the Carlqvist relation, and if it is replaced by the more general Bennett relation, then the designated regions of the chart are not valid. Carlqvist further notes that by using

8025-473: The material and decreases with an increase in density. For ideal gases, the bulk modulus K is simply the gas pressure multiplied by the dimensionless adiabatic index , which is about 1.4 for air under normal conditions of pressure and temperature. For general equations of state , if classical mechanics is used, the speed of sound c can be derived as follows: Consider the sound wave propagating at speed v {\displaystyle v} through

8132-563: The medium perpendicular to the direction of wave travel; the direction of shear-deformation is called the " polarization " of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations. These different waves (compression waves and the different polarizations of shear waves) may have different speeds at the same frequency. Therefore, they arrive at an observer at different times, an extreme example being an earthquake , where sharp compression waves arrive first and rocking transverse waves seconds later. The speed of

8239-451: The medium through which a sound wave is propagating. At 0 °C (32 °F), the speed of sound in air is about 331 m/s (1,086 ft/s; 1,192 km/h; 740 mph; 643 kn). The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in ordinary air, deviating slightly from ideal behavior. In colloquial speech, speed of sound refers to

8346-407: The metal or glass to spall off and enter the fuel, rapidly cooling the plasma. Unless the plasma could be made stable, this loss process would make fusion impossible. In the mid-1950s, two possible solutions appeared. In the fast-pinch concept, a linear device would undergo the pinch so quickly that the plasma as a whole would not move, instead only the outermost layer would begin to pinch, creating

8453-406: The middle of the mug. When current is applied, it begins to arc at the path of least resistance, at the end near the insulator disk. This causes the gas in the area to rapidly ionize, and current begins to flow through it to the outer electrode. The current creates a magnetic field that begins to push the plasma down the tube towards the open end. It reaches the end in microseconds. When it reaches

8560-547: The plasma focus experiments have been extended to sub-kilojoules devices and the scales rules have been stretched up to region less than one joule. Their studies have contributes to know that is possible to scale the plasma focus in a wide range of energies and sizes keeping the same value of ion density, magnetic field, plasma sheath velocity, Alfvén speed and the quantity of energy per particle. Therefore, fusion reactions are even possible to be obtained in ultraminiature devices (driven by generators of 0.1J for example), as they are in

8667-402: The plasma. Another advantage claimed is the capability of direct conversion of the energy of the fusion products into electricity, with an efficiency potentially above 70%. Experiments and computer simulations to investigate the capability of DPF for fusion power are underway at Lawrenceville Plasma Physics (LPP) under the direction of Eric Lerner , who explained his "Focus Fusion" approach in

8774-432: The properties of the substance through which the wave is travelling. In solids, the speed of transverse (or shear) waves depends on the shear deformation under shear stress (called the shear modulus ), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on compressibility . In fluids, only the medium's compressibility and density are

8881-596: The relations above, and a derivative, it is possible to describe the Bennett pinch, the Jeans criterion (for gravitational instability, in one and two dimensions), force-free magnetic fields , gravitationally balanced magnetic pressures, and continuous transitions between these states. A fictionalized pinch-generating device was used in Ocean's Eleven , where it was used to disrupt Las Vegas's power grid just long enough for

8988-407: The same 60 kJ energy input. In addition, mean ion energy increased to a record of 240 ± 20 keV for any confined fusion plasma. A deuterium-nitrogen mix and corona-discharge pre-ionization reduced the fusion yield standard deviation by 4x to about 15%. In 2019, the team conducted a series of experiments replacing tungsten electrodes with beryllium electrodes (termed Focus Fusion 2B). After 44 shots,

9095-1421: The same at the two ends of the tube, therefore the mass flux j = ρ v {\displaystyle j=\rho v} is constant and v d ρ = − ρ d v {\displaystyle v\,d\rho =-\rho \,dv} . Per Newton's second law , the pressure-gradient force provides the acceleration: d v d t = − 1 ρ d P d x → d P = ( − ρ d v ) d x d t = ( v d ρ ) v → v 2 ≡ c 2 = d P d ρ {\displaystyle {\begin{aligned}{\frac {dv}{dt}}&=-{\frac {1}{\rho }}{\frac {dP}{dx}}\\[1ex]\rightarrow dP&=(-\rho \,dv){\frac {dx}{dt}}=(v\,d\rho )v\\[1ex]\rightarrow v^{2}&\equiv c^{2}={\frac {dP}{d\rho }}\end{aligned}}} And therefore: c = ( ∂ P ∂ ρ ) s = K s ρ , {\displaystyle c={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {K_{s}}{\rho }}},} If relativistic effects are important,

9202-417: The same plasma characteristics and the same radiation products. This is due to the scalability of plasma phenomena. See also plasmoid , the self-contained magnetic plasma ball that may be produced by a dense plasma focus. The fact that the plasma energy density is constant throughout the range of plasma focus devices, from big to small, is related to the value of a design parameter that needs to be kept at

9309-466: The section on gases in specific heat capacity for a more complete discussion of this phenomenon. For air, we introduce the shorthand R ∗ = R / M a i r . {\displaystyle R_{*}=R/M_{\mathrm {air} }.} In addition, we switch to the Celsius temperature θ = T − 273.15 K , which is useful to calculate air speed in

9416-426: The sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for c air have been found to vary slightly from experimentally determined values. Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic . His result

9523-404: The speed of sound increases with density. This notion is illustrated by presenting data for three materials, such as air, water, and steel and noting that the speed of sound is higher in the denser materials. But the example fails to take into account that the materials have vastly different compressibility, which more than makes up for the differences in density, which would slow wave speeds in

9630-423: The speed of sound is about 75% of the mean speed that the atoms move in that gas. For a given ideal gas the molecular composition is fixed, and thus the speed of sound depends only on its temperature . At a constant temperature, the gas pressure has no effect on the speed of sound, since the density will increase, and since pressure and density (also proportional to pressure) have equal but opposite effects on

9737-506: The speed of sound is calculated from the relativistic Euler equations . In a non-dispersive medium , the speed of sound is independent of sound frequency , so the speeds of energy transport and sound propagation are the same for all frequencies. Air, a mixture of oxygen and nitrogen, constitutes a non-dispersive medium. However, air does contain a small amount of CO 2 which is a dispersive medium, and causes dispersion to air at ultrasonic frequencies (greater than 28  kHz ). In

9844-404: The speed of sound is dependent solely upon temperature; see § Details below. In such an ideal case, the effects of decreased density and decreased pressure of altitude cancel each other out, save for the residual effect of temperature. Since temperature (and thus the speed of sound) decreases with increasing altitude up to 11 km , sound is refracted upward, away from listeners on

9951-539: The speed of sound waves in air . However, the speed of sound varies from substance to substance: typically, sound travels most slowly in gases , faster in liquids , and fastest in solids . For example, while sound travels at 343 m/s in air, it travels at 1481 m/s in water (almost 4.3 times as fast) and at 5120 m/s in iron (almost 15 times as fast). In an exceptionally stiff material such as diamond, sound travels at 12,000 m/s (39,370 ft/s),  – about 35 times its speed in air and about

10058-490: The speed of sound, and the two contributions cancel out exactly. In a similar way, compression waves in solids depend both on compressibility and density—just as in liquids—but in gases the density contributes to the compressibility in such a way that some part of each attribute factors out, leaving only a dependence on temperature, molecular weight, and heat capacity ratio which can be independently derived from temperature and molecular composition (see derivations below). Thus, for

10165-402: The spheres represent the material's molecules and the springs represent the bonds between them. Sound passes through the system by compressing and expanding the springs, transmitting the acoustic energy to neighboring spheres. This helps transmit the energy in-turn to the neighboring sphere's springs (bonds), and so on. The speed of sound through the model depends on the stiffness /rigidity of

10272-429: The temperature and molecular weight, thus making only the completely independent properties of temperature and molecular structure important (heat capacity ratio may be determined by temperature and molecular structure, but simple molecular weight is not sufficient to determine it). Sound propagates faster in low molecular weight gases such as helium than it does in heavier gases such as xenon . For monatomic gases,

10379-538: The use of plasma focus devices as plasma accelerators for studies of materials under intense fusion-relevant pulses. In addition, Chilean Nuclear Energy Commission currently operates the facility SPEED-2, the largest Plasma Focus facility of the southern hemisphere. Since the beginning of 2009, a number of new plasma focus machines have been/are being commissioned including the INTI Plasma Focus in Malaysia,

10486-632: The well-known neutron saturation effect is better correlated to a scaling deterioration mechanism. This is due to the increasing dominance of the axial phase dynamic resistance as capacitor bank impedance decreases with increasing bank energy (capacitance). In principle, the resistive saturation could be overcome by operating the pulse power system at a higher voltage. The International Centre for Dense Magnetised Plasmas (ICDMP) in Warsaw Poland, operates several plasma focus machines for an international research and training programme. Among these machines

10593-751: The whole electromagnetic spectrum including radio waves , microwaves , infrared , x-rays , gamma rays , synchrotron radiation , and visible light . They also produce neutrons , as a product of fusion. Pinches are used to generate X-rays and the intense magnetic fields generated are used in electromagnetic forming of metals. They also have applications in particle beams including particle beam weapons , astrophysics studies and it has been proposed to use them in space propulsion. A number of large pinch machines have been built to study fusion power ; here are several: Many high-voltage electronics enthusiasts make their own crude electromagnetic forming devices. They use pulsed power techniques to produce

10700-655: The widely differing performance of different machines. Thus to result in the necessary plasma energy density (which is found to be a near constant for optimized neutron production) requires widely differing initial storage density. A network of ten identical DPF machines operates in eight countries around the world. This network produces research papers on topics including machine optimization & diagnostics (soft X-rays, neutrons, electron and ion beams), applications (microlithography, micromachining, materials modification and fabrication, imaging & medical, astrophysical simulation) as well as modeling & computation. The network

10807-566: The world's first controlled thermonuclear fusion experiment was accomplished using a theta-pinch machine named Scylla I at the Los Alamos National Laboratory . A cylinder full of deuterium was converted into a plasma and compressed to 15 million degrees Celsius under a theta-pinch effect. Lastly, at Imperial College in 1960, led by R Latham, the Plateau–Rayleigh instability was shown, and its growth rate measured in

10914-422: Was 325 mm . This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as 304.8 mm , making the speed of sound at 20 °C (68 °F) 1,055 Parisian feet per second). Derham used a telescope from the tower of the church of St. Laurence, Upminster to observe the flash of a distant shotgun being fired, and then measured the time until he heard

11021-419: Was able to develop for the first time design maps combining geometrical and operational parameters, showing that there is always an optimum gun length and charging pressure which maximize the neutron emission. Currently there is a complete finite-elements code validated against numerous experiments, which can be used confidently as a design tool for Plasma Focus. In Chile, at the Chilean Nuclear Energy Commission

11128-474: Was founded on 25 February 2008 to promote correct and innovative use of the Lee Model code and to encourage the application of plasma focus numerical experiments. IPFS research has already extended numerically-derived neutron scaling laws to multi-megajoule experiments. These await verification. Numerical experiments with the code have also resulted in the compilation of a global scaling law indicating that

11235-435: Was missing the factor of γ but was otherwise correct. Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of γ = 1.4000 requires that the gas exists in a temperature range high enough that rotational heat capacity

11342-527: Was organized by Sing Lee in 1986 and is coordinated by the Asian African Association for Plasma Training, AAAPT . A simulation package, the Lee Model, has been developed for this network but is applicable to all plasma focus devices. The code typically produces excellent agreement between computed and measured results, and is available for downloading as a Universal Plasma Focus Laboratory Facility. The Institute for Plasma Focus Studies IPFS

11449-411: Was the publication in 1934 of an analysis of the radial pressure balance in a static Z-pinch by Bennett (see the following section for details). Thereafter, the experimental and theoretical progress on pinches was driven by fusion power research. In their article on the "Wire-array Z-pinch: a powerful x-ray source for ICF ", M G Haines et al. , wrote on the "Early history of Z-pinches". In 1958,

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