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Critical density

The term critical density has distinct meanings in various fields of physics. In cosmology, it refers to the theoretical density of matter and energy required for a flat universe. In thermodynamics, it denotes the density of a fluid at its critical point, where the distinction between liquid and gas phases vanishes, enabling unique properties like vanishing surface tension.^[1] In plasma physics, it is the electron density at which the plasma frequency equals the frequency of an electromagnetic wave, acting as a cutoff for wave propagation in the medium.#Definition) Details on these contexts are covered in subsequent sections. In cosmology, the critical density is the theoretical threshold density of matter and energy in the universe that would result in a flat spatial geometry, balancing the effects of gravitational attraction and the initial expansion momentum as described by general relativity. It is given by the formula \rho_c = \frac{3H^2}{8\pi [G](/page/Gravitational_constant)}, where H is the Hubble parameter measuring the current expansion rate, and G is the gravitational constant; using the Planck-measured H \approx 67.4 km/s/Mpc (as of 2018), this yields a present-day value of approximately $8.6 \times 10^{-27} kg/m³, equivalent to roughly five hydrogen atoms per cubic meter.^[2]^[3]^[4] The concept arises from the Friedmann equations, which govern the dynamics of a homogeneous and isotropic universe, and the critical density serves as a benchmark for classifying the universe's large-scale geometry and ultimate fate. The density parameter \Omega = \frac{\rho}{\rho_c} quantifies how the actual total energy density \rho (including ordinary matter, dark matter, radiation, and dark energy) compares to this threshold: \Omega > 1 implies a closed universe with positive spatial curvature that would eventually recollapse; \Omega = 1 a flat universe with zero curvature and perpetual expansion (decelerating without dark energy); and \Omega < 1 an open universe with negative curvature and eternal expansion.^[5]^[6]^[7] In the standard \LambdaCDM model, dark energy (modeled as a cosmological constant \Lambda) dominates the energy budget, driving accelerated expansion despite the total density being near critical; observations from the cosmic microwave background (CMB) by the Planck satellite confirm a nearly flat geometry with \Omega_K = -0.012 \pm 0.010 (where \Omega_K = 1 - \Omega_\mathrm{total}), \Omega_m \approx 0.315 \pm 0.007 for matter (baryonic plus dark), and \Omega_\Lambda \approx 0.685 \pm 0.007 for dark energy, summing to \Omega_\mathrm{total} \approx 1 within measurement precision (as of 2024).^[8] This near-critical density has profound implications, supporting an infinite, ever-expanding universe while resolving tensions in early predictions of lower matter densities from visible light alone. Recent surveys like DESI reinforce these findings, with \Omega_m \approx 0.298 \pm 0.009 (as of 2024), though they hint at potential evolution in dark energy density at low redshifts.^[9]

In cosmology

Definition and physical significance

In cosmology, the critical density refers to the hypothetical average density of matter and energy in the universe that would result in a flat spatial geometry, where the inward gravitational attraction precisely balances the outward force of expansion, leading to an eternal decelerating expansion without eventual recollapse or indefinite acceleration.^[10] This density serves as a benchmark for understanding the universe's overall energy content and its long-term evolution.^[4] The physical significance of the critical density lies in its role as a dividing line that determines the geometry and ultimate fate of the universe, as described within the framework of the Friedmann equations. If the actual density exceeds the critical value (Ω > 1, where Ω is the density parameter), the universe is closed and will eventually recollapse under gravity; if below (Ω < 1), it is open and will expand forever; at exactly the critical density (Ω = 1), the universe is flat and expands indefinitely, approaching zero expansion rate only after infinite time.^[11] This parameter not only influences cosmic curvature but also connects to broader implications for structure formation and the balance between matter, radiation, and dark energy components.^[12] The concept of critical density emerged in the early development of the Big Bang model, first introduced by Alexander Friedmann in his 1922 derivation of expanding universe solutions from Einstein's general relativity equations, and independently by Georges Lemaître in 1927, who emphasized the dynamic evolution of cosmic density. In the modern context, inflationary theory, proposed by Alan Guth in 1981, gains relevance by predicting that the universe's density should be extremely close to the critical value due to the rapid exponential expansion in the early universe, which flattens spatial curvature regardless of initial conditions.^[13] Critical density is typically expressed in units of kilograms per cubic meter (kg/m³) or, more commonly in cosmological studies, as a dimensionless fraction of the total energy density through the parameter Ω.^[4]

Mathematical derivation

The mathematical derivation of the critical density in cosmology originates from the first Friedmann equation, which arises from applying the Einstein field equations to a homogeneous and isotropic universe modeled by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric.^[14] This equation relates the expansion rate of the universe to its energy content, curvature, and the cosmological constant.^[10] The first Friedmann equation is given by

H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G}{3} \rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3},

where H is the Hubble parameter, a(t) is the scale factor describing the relative expansion of the universe, \rho is the total matter and radiation energy density, k is the spatial curvature parameter (+1 for closed, $0 for flat, and -1 for open geometries), G is the gravitational constant, c is the speed of light, and \Lambda is the cosmological constant. To derive the critical density, consider the simplified case of a flat universe (k = 0) without a cosmological constant (\Lambda = 0). The equation then reduces to

H^2 = \frac{8\pi G}{3} \rho,

which can be rearranged to solve for the density:

\rho_c = \frac{3 H^2}{8 \pi G}.

This \rho_c represents the critical density, the value at which the universe is spatially flat and expands forever at a decelerating rate.^[10] In the more general case including the cosmological constant, the critical density retains the same form \rho_c = 3 H^2 / (8 \pi G), but now applies to the total effective energy density, where the cosmological constant contributes an equivalent vacuum energy density \rho_\Lambda = \Lambda / (8 \pi G).^[14] The total density parameter is then \Omega = \rho / \rho_c, with contributions from matter (\Omega_m), radiation (\Omega_r), and dark energy (\Omega_\Lambda); a flat universe requires \Omega = 1. For the geometry without dark energy, \Omega > 1 implies a closed universe (k = +1), while \Omega < 1 implies an open universe (k = -1).^[10] This derivation assumes a homogeneous and isotropic universe governed by the FLRW metric, neglecting quantum gravitational effects and specific initial conditions such as those from inflation.

Current estimates and observations

As of the Planck 2018 analysis, the estimate of the critical density is ρ_c ≈ 8.5 × 10^{-27} kg m^{-3}, derived from the Hubble constant H_0 = 67.4 ± 0.5 km s^{-1} Mpc^{-1} measured using cosmic microwave background (CMB) anisotropies.^[15] In natural units, this corresponds to approximately 4.8 × 10^{-6} GeV cm^{-3}.^[16] This value reflects the density required for a flat universe under the Friedmann equation, with measurements from CMB, baryon acoustic oscillations (BAO), and other probes consistently placing the total density parameter near unity: Ω_total ≈ 1.00 ± 0.02.^[15] Note that ρ_c scales as H_0^2, and the ongoing Hubble tension—with local measurements yielding H_0 ≈ 70–73 km/s/Mpc—implies a higher ρ_c in those contexts. The breakdown of density parameters in the standard ΛCDM model shows matter (baryonic and dark) contributing Ω_m ≈ 0.315 ± 0.007 (Planck 2018), dark energy dominating with Ω_Λ ≈ 0.685 ± 0.007, and radiation negligible today (Ω_r ≪ 10^{-4}). Recent combined constraints as of 2025, including from DESI, refine Ω_m to ≈ 0.30 ± 0.01. Near-flatness is confirmed by the angular scale of CMB anisotropies, θ ≈ 1°, which matches predictions for a flat geometry and supports spatial flatness to within 0.2% precision when combined with BAO data.^[15] Key observations underpinning these estimates include CMB measurements from the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck, which tightly constrain Ω_total through temperature and polarization power spectra.^[15] Type Ia supernova data, such as from the Pantheon sample, provide evidence for cosmic acceleration and Ω_Λ > 0 by revealing dimmer distant supernovae than expected in a matter-dominated universe. BAO features in galaxy surveys like the Sloan Digital Sky Survey (SDSS) refine the expansion history H(z) and corroborate Ω_m and H_0 values when combined with CMB results.^[15] The Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2, 2025) provides updated BAO measurements from over 14 million galaxies and quasars, supporting near-critical total density but hinting at evolving dark energy with a 4.2σ preference over a constant cosmological constant in some models. These results suggest potential time-variation in dark energy density at low redshifts, with Ω_m ≈ 0.30 ± 0.01 in ΛCDM.^[17]^[18] Early data from the Euclid satellite (launched 2023) and the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST, first light 2025) further tighten constraints on curvature and dark energy, aiming for 1% precision. Uncertainties persist, notably the Hubble tension between CMB-derived H_0 ≈ 67.4 km s^{-1} Mpc^{-1} and local measurements from Cepheid-calibrated supernovae yielding H_0 ≈ 70–73 km s^{-1} Mpc^{-1}, implying a 4-6σ discrepancy that affects ρ_c precision. Future missions are projected to refine these parameters to 1% accuracy, potentially resolving tensions. Historically, early estimates assumed a matter-dominated universe with Ω_m < 1 until the 1998 discovery of cosmic acceleration from Type Ia supernovae, which prompted the adoption of the ΛCDM model incorporating dark energy and near-critical total density.

In thermodynamics

Critical point fundamentals

In thermodynamics, the critical point marks the endpoint of the liquid-vapor coexistence curve on a phase diagram, where the distinction between the liquid and vapor phases ceases to exist, and the fluid exhibits properties intermediate between those of a liquid and a gas. At this point, the critical temperature T_c, critical pressure P_c, and critical density \rho_c are uniquely defined such that the meniscus between phases disappears, and the fluid becomes a supercritical fluid above T_c. For water, these parameters are T_c = 647 K and P_c = 22.06 MPa. Key thermodynamic properties diverge at the critical point, signaling the onset of critical phenomena. The isothermal compressibility \kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T becomes infinite, reflecting extreme susceptibility to volume changes under pressure. Similarly, the specific heat at constant volume C_V diverges, indicating large energy fluctuations, while the correlation length \xi, which measures the spatial extent of density fluctuations, also diverges as \xi \to \infty. These divergences underpin scaling laws that describe the behavior of fluids near criticality, unifying diverse physical responses through universal exponents.^[19] The empirical discovery of the critical point traces back to 1822, when Charles Cagniard de la Tour observed the disappearance of the liquid-vapor interface in sealed tubes containing fluids like alcohol and ether under heating, using a modified Papin's digester to reach high pressures.^[20] Dmitri Mendeleev further explored these phenomena in the 1860s, naming the "critical point" and emphasizing its role in the continuity between gaseous and liquid states.^[21] Theoretical insight arrived in 1873 with Johannes Diderik van der Waals' doctoral thesis, which introduced an equation of state incorporating intermolecular attractions and finite molecular volume to explain phase transitions. The van der Waals equation is given by

P = \frac{RT}{V_m - b} - \frac{a}{V_m^2},

where V_m is the molar volume, R is the gas constant, and a and b account for attractive forces and excluded volume, respectively. To locate the critical point, the conditions for an inflection point in the isotherm are applied:

\left( \frac{\partial P}{\partial V_m} \right)_T = 0, \quad \left( \frac{\partial^2 P}{\partial V_m^2} \right)_T = 0.

Solving these yields the critical parameters V_c = 3b, P_c = \frac{a}{27b^2}, and T_c = \frac{8a}{27Rb}, providing a foundational model for predicting critical behavior despite its simplifications.^[22]

Critical density properties

The critical density, denoted as \rho_c, represents the mass density of a pure substance at its critical point, where the coexisting liquid and vapor phases have identical properties and the phase boundary vanishes. At this point, the densities of the liquid and vapor become equal, marking the transition to a supercritical fluid state. For example, water has a critical density of approximately 322 kg/m³, while carbon dioxide exhibits a value of about 468 kg/m³.^[23]^[24] Theoretical calculations of \rho_c can be derived from equations of state such as the van der Waals model, which predicts the critical molar volume V_{m,c} = 3b, where b is the excluded volume parameter per mole; thus, \rho_c = \frac{M}{3b}, with M being the molar mass. Experimentally, \rho_c is determined by heating a fluid sample in a sealed capillary tube until the meniscus between liquid and vapor phases disappears, indicating the critical isotherm. Advanced setups, such as piston-cylinder apparatuses, allow precise control of pressure and temperature to measure density via volume changes at the critical conditions.^[25]^[26] Near the critical point, fluids display universal behavior when expressed in reduced variables, such as the reduced temperature T_r = T / T_c and reduced density \rho_r = \rho / \rho_c, where T_c is the critical temperature; this scaling reveals similar thermodynamic paths across substances, as described by the principle of corresponding states. A key manifestation is the law of rectilinear diameters, which states that the average density along the liquid-vapor coexistence curve, (\rho_l + \rho_v)/2, varies linearly with temperature: (\rho_l + \rho_v)/2 = a - bT, where a and b are constants fitted to experimental data./16%3A_The_Properties_of_Gases/16.04%3A_The_Law_of_Corresponding_States)^[27] The value of \rho_c is primarily influenced by molecular size and repulsive interactions, encapsulated in the van der Waals parameter b, which accounts for the finite volume of molecules; larger molecules yield smaller \rho_c due to increased excluded volume. Attractive interactions, governed by the parameter a, play a secondary role by modulating the critical temperature but indirectly affect \rho_c through overall molecular packing; polar molecules, with stronger attractions, often exhibit \rho_c values shaped by both factors compared to nonpolar counterparts.^[28]

Applications in phase transitions

Supercritical fluids exist above the critical temperature T_c and critical pressure P_c, where the distinction between liquid and gas phases vanishes, resulting in a state with gas-like diffusivity and liquid-like solvating power, particularly when the density approaches the critical density \rho_c. This unique combination enables efficient mass transfer in processes like extraction, where supercritical carbon dioxide (CO₂) is commonly employed due to its mild T_c = 31.1^\circC and P_c = 73.8 bar, with \rho_c \approx 468 kg/m³. For instance, in decaffeination of green coffee beans, supercritical CO₂ at densities of approximately 0.5–0.8 \rho_c (achieved at pressures of 200–350 bar and temperatures of 40–60°C) selectively extracts caffeine while preserving flavor compounds, yielding high-purity extracts without residual solvents.^[29]^[30]^[31] In studies of phase transitions, critical density serves as a key parameter for modeling phenomena near the critical point, such as opalescence, where diverging density fluctuations scatter light intensely, leading to a milky appearance in fluids like sulfur hexafluoride (SF₆). These fluctuations, characterized by the correlation length diverging as \xi \propto |T - T_c|^{-\nu} (with \nu \approx 0.63 in the 3D Ising universality class), align the critical behavior of fluid phase transitions with the Ising model, where order parameter jumps (e.g., in density for liquid-gas transitions) mirror magnetization changes. For binary fluid mixtures, critical density informs the location of consolute points, where upper or lower miscibility gaps close; for example, in isobutyric acid-water mixtures, the critical composition and density shifts along the coexistence curve help predict demixing under gravity or temperature gradients.^[32]^[33]^[34] Industrial applications leverage supercritical states near \rho_c for enhanced processes involving phase changes. In enhanced oil recovery (EOR), supercritical CO₂ injection at densities comparable to liquids (around 0.6–0.8 g/cm³) reduces oil viscosity and interfacial tension, improving sweep efficiency in reservoirs; field projects demonstrate recovery increases of 10–20% over conventional methods by exploiting CO₂'s ability to penetrate pore spaces without phase separation. Similarly, supercritical water oxidation (SCWO) treats hazardous wastes at T_c = 647 K and \rho_c \approx 322 kg/m³, where organic compounds achieve complete miscibility and rapid oxidation to CO₂ and H₂O, achieving >99% destruction efficiency for pollutants like polychlorinated biphenyls in minutes, far surpassing subcritical methods.^[35]^[36]^[37] Research implications of critical density extend to understanding nucleation and cavitation dynamics, where near-critical conditions amplify fluctuation-driven bubble formation; molecular simulations reveal critical states during cavitation inception in water, with system density and pressure stabilizing transient nuclei before collapse or growth. In astrophysics, the helium critical point (T_c \approx 5.2 K, \rho_c \approx 69 kg/m³ under classical conditions) informs models of phase transitions in stellar interiors, particularly in degenerate helium layers of white dwarfs or giant planet cores, where high pressures shift the liquid-gas boundary and influence convective stability during helium flashes.^[38]^[39] Challenges in these applications include the high pressures required to maintain densities near \rho_c, often exceeding 100 bar for CO₂ or 220 bar for water, which demand robust materials and increase operational costs. Additionally, scaling laws for supercritical flows, such as those predicting heat transfer deterioration due to buoyancy-induced density gradients, complicate extrapolation from lab to industrial scales, with non-dimensional parameters like the Rayleigh number highlighting deviations from ideal mixing.^[40]^[41]

In plasma physics

Plasma frequency and critical density

In plasma physics, the plasma frequency represents the natural oscillation frequency of electrons in response to perturbations in a plasma. It arises from the collective motion of free electrons under the restoring force of the electric field generated by charge separation. The angular plasma frequency \omega_p is given by

\omega_p = \sqrt{\frac{n_e e^2}{\epsilon_0 m_e}},

where n_e is the electron number density, e is the elementary charge, m_e is the electron mass, and \epsilon_0 is the vacuum permittivity.^[42] This frequency sets a fundamental timescale for electron dynamics, with oscillations occurring at \omega_p when the plasma is unmagnetized and collisionless. The critical plasma density n_c is defined as the electron density at which the plasma frequency equals the angular frequency \omega of an incident electromagnetic wave, i.e., \omega_p = \omega. Above this density, the wave cannot propagate through the plasma. The expression for n_c is

n_c = \frac{\epsilon_0 m_e \omega^2}{e^2}.

For electromagnetic waves such as light, \omega = 2\pi [c](/page/Speed_of_light) / \lambda, where c is the speed of light and \lambda is the wavelength, making n_c dependent on the wave's frequency.^[43] This threshold emerges from the dispersion relation for electromagnetic waves in the cold plasma approximation, which neglects thermal effects:

\omega^2 = \omega_p^2 + c^2 k^2,

where k is the wave number. For real k (propagating waves), \omega > \omega_p is required; otherwise, when n_e > n_c (so \omega_p > \omega), k becomes imaginary, rendering the wave evanescent and leading to reflection at the plasma boundary. The refractive index \eta = c k / \omega = \sqrt{1 - \omega_p^2 / \omega^2} becomes imaginary above n_c, confirming the cutoff.^[44] In ionospheric physics, where radio waves interact with the partially ionized upper atmosphere, n_c \approx 10^{12} m^{-3} for typical frequencies around 10 MHz, determining the reflection of high-frequency signals for long-distance communication. In fusion plasmas, higher-frequency lasers (e.g., near-infrared with \lambda \approx 1 \mum, corresponding to \omega \approx 2 \times 10^{15} rad/s) yield n_c on the order of $10^{27} m^{-3}, a key parameter for laser-plasma interactions in inertial confinement fusion experiments.^[45]^[46] The relation between plasma frequency and critical density was derived in the early 20th century by Hendrik Lorentz as part of his electron theory, applied to explain radio wave propagation in the ionosphere.^[47]

Role in wave propagation

In plasma physics, the critical density n_c plays a pivotal role in determining the propagation characteristics of electromagnetic waves, primarily through the cutoff phenomenon associated with the plasma frequency \omega_p. When the electron density n_e exceeds n_c, corresponding to \omega_p > \omega where \omega is the wave frequency, the wave becomes evanescent and is reflected at the plasma boundary, preventing penetration into the denser region. This reflection arises because the refractive index becomes imaginary, leading to total reflection without energy transmission. Conversely, for n_e < n_c (\omega_p < \omega), waves can propagate, exhibiting a phase velocity greater than the speed of light c but a group velocity less than c, ensuring that energy transport remains subluminal. A prominent application of this cutoff is the radio blackout experienced by re-entry vehicles, where the high-temperature plasma sheath formed during atmospheric re-entry has n_e > n_c for typical communication frequencies (e.g., VHF or X-band), reflecting radio signals and disrupting telemetry.^[48]^[49]^[50] In magnetized plasmas, the critical density also influences resonant phenomena, such as the upper hybrid resonance, which occurs at the frequency \omega_{uh} = \sqrt{\omega_p^2 + \omega_c^2}, where \omega_c is the electron cyclotron frequency. At this resonance, electromagnetic waves can couple strongly to electrostatic oscillations, enhancing absorption or scattering near regions where n_e \approx n_c. This effect is particularly relevant for wave-plasma interactions in controlled environments, where it facilitates energy transfer or diagnostic insights by observing frequency-dependent cutoffs.^[51]^[52] In inhomogeneous plasmas, where density gradients exist near n_c, advanced propagation effects emerge, including wave tunneling and mode conversion. Tunneling allows partial transmission of waves through evanescent regions via quantum-like barrier penetration, while mode conversion transforms electromagnetic modes into electrostatic ones (e.g., Langmuir waves) at turning points defined by n_c. These processes are crucial in astrophysical contexts, such as the solar corona, where the critical density sets the cutoff for radio emission propagation; waves generated at plasma frequencies below the local \omega_p (corresponding to n_e > n_c for the observing frequency) are trapped, limiting observable radio bursts to heights where n_e \leq n_c. In inertial confinement fusion (ICF), the critical density layer (n_c \approx 10^{27} m^{-3} for 1 μm lasers) marks the point of intense laser absorption via inverse bremsstrahlung, driving plasma ablation and implosion; disruptions here, such as instabilities, can reduce energy coupling efficiency and degrade fusion yield.^[53]^[54]^[55]^[56]^[57] These propagation behaviors assume a collisionless plasma, where damping is minimal. However, collisions introduce additional attenuation through collisional damping, which broadens resonances and reduces wave amplitude near n_c, while collisionless mechanisms like Landau damping further dissipate energy in the underdense regime. Such effects must be accounted for in high-density or warm plasmas to accurately model wave access and absorption.^[58]^[59]^[60]

Experimental measurements

Several experimental techniques are employed to measure the critical plasma density (n_c), which corresponds to the point where electromagnetic waves are reflected due to the plasma frequency matching the wave frequency. Microwave interferometry is a widely used non-invasive method that detects phase shifts in microwave signals propagating through the plasma, where the phase shift is proportional to the electron density (n_e), allowing inference of n_c through observation of cutoff frequencies.^[61]^[62] This technique has been applied in various low-to-moderate density plasmas, with limitations in high-density regimes above $10^{20} m^{-3} due to signal attenuation.^[63] Thomson scattering provides a direct, laser-based diagnostic by scattering probe light off plasma electrons, measuring the Doppler-broadened spectrum to infer electron temperature and density distributions near n_c.^[64]^[65] In collective Thomson scattering regimes, it resolves velocity distributions in underdense plasmas approaching critical conditions, enabling precise n_e mapping without significant perturbation.^[66] Reflectometry complements these by launching microwaves into the plasma and detecting reflections at the critical density layer, where the cutoff occurs, providing profile information through frequency sweeps.^[67]^[68] This method is particularly effective for real-time density fluctuation studies in fusion devices.^[69] In laboratory settings, tokamaks such as the Joint European Torus (JET) utilize Langmuir probes to measure local n_e up to $10^{20} m^{-3} in the scrape-off layer and core, where probe current-voltage characteristics yield density and temperature data essential for identifying n_c thresholds.^[70]^[71] These invasive probes are calibrated against non-perturbing methods like interferometry for accuracy in high-field environments. At the National Ignition Facility (NIF), laser-plasma experiments probe n_c via X-ray imaging of absorption edges and emission profiles in underdense targets, revealing density gradients critical for inertial confinement fusion.^[72]^[73] Natural plasma environments are observed using ionosondes, which transmit vertical radio pulses and measure reflections from the F-layer, yielding n_c \approx 10^{11}--$10^{12} m^{-3} based on the critical frequency f_oF_2.^[74]^[75] Solar radio bursts provide another remote diagnostic, with dynamic spectra showing cutoff frequencies in type III bursts that trace electron beam propagation through coronal plasmas, inferring local n_c from frequency drifts.^[76]^[77] Measuring n_c faces challenges including calibration for temperature-dependent refractive index variations, which can bias phase-based techniques, and limited spatial resolution in dense plasmas where gradients near n_c are steep.^[78] Magnetic fields further complicate inferences by altering the effective cutoff through cyclotron effects, introducing errors up to 20% in magnetized plasmas without corrections.^[79] As of 2025, advances with petawatt lasers have enabled direct probing of n_c in underdense plasmas for inertial fusion, using techniques like betatron radiation and interferometry to map relativistic electron dynamics at near-critical densities.^[80]^[81]

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Apr 25, 2024 · Critical opalescence occurs at this singularity. Discovered in 1822, it is known to be driven by diverging fluctuations in the density. During ...
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Apparent mean-field criticality of liquid–liquid phase transitions in ...
It is now common knowledge that the critical properties of phase transitions of fluids and fluid mixtures agree with that of the 3d-Ising model.
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Densities of binary liquid mixtures near their consolute points
The predicted tangency of the critical isotherm and coexistence curve is confirmed for isobutyric acid + water within the uncertainties in the estimates.
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A review of the supercritical CO2 fluid applications for improved oil ...
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[PDF] Carbon Dioxide Enhanced Oil Recovery
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Applications of Supercritical Water in Waste Treatment and ... - MDPI
Supercritical water oxidation (SCWO) is a waste treatment method that can convert hazardous organics completely and quickly into water and carbon dioxide [2].
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A study of cavitation nucleation in pure water using molecular ...
The results show that critical states exist in the process of cavitation nucleation. In the critical state, the energy, density, and pressure of the system ...
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Fluid helium at conditions of giant planetary interiors - PNAS
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Feb 25, 2025 · Despite achievements made so far, challenges persist in terms of high-fidelity modeling and simulations. Quantitative experimental data are ...
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Plasma Frequency
There is a different plasma frequency for each species. However, the relatively large electron frequency is, by far, the most important of these, and references ...Missing: critical | Show results with:critical
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critical_density — PlasmaPy 2025.10.0 documentation
The critical density for a given frequency of radiation is defined as the value at which the electron plasma frequency equals the frequency of the radiation.
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Waves in Unmagnetized Plasmas - Richard Fitzpatrick
Waves in Unmagnetized Plasmas. Let us now investigate the cold-plasma dispersion relation in detail. It is instructive to first consider the limit in which ...
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IMPC Website: Electron Density
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Plasma Frequency - an overview | ScienceDirect Topics
Plasma frequency is defined as the lowest frequency at which an electromagnetic wave can propagate through a plasma without being strongly absorbed, ...
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[PDF] The Lorentz Oscillator and its Applications - MIT OpenCourseWare
The plasma frequency for “non-plasma” materials stands for the natural collective oscillation frequency of the “sea” of electrons in the material, not of ...
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[PDF] Chapter 5 Electromagnetic Waves in Plasmas
This is called the 'Plasma Frequency' (more properly ωpe the 'electron' plasma frequency). If we allow for ion motions we get an ion conductivity iniq2 σi ...
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[PDF] Causes and Mitigation of Radio Frequency (RF) Blackout During ...
Jan 26, 2007 · During flight trajectories, vehicles can generate electron densities with plasma frequencies exceeding the RF transmission frequency, leading ...
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[PDF] Radio blackout alleviation and plasma diagnostic results from a ...
The X-band (9210 MHz) telemetry signal was received down to 144 000 ft (43.9 km), approximately 120 000 ft (36.6 km) after VHF blackout.
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Upper hybrid wave excitation and particle acceleration by resonant ...
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[PDF] Plasma waves in the fluid picture II
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Evidence for Langmuir wave tunneling in the inhomogeneous solar ...
Oct 23, 2002 · Incident Langmuir waves which are nearly aligned with the density gradient are also partially converted to transverse electromagnetic waves at ...
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[PDF] Mode Conversion Of Langmuir To Electromagnetic Waves With ...
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[PDF] Plasma Waves and Instabilities - Princeton University
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A future of inertial confinement fusion without laser-plasma instabilities
May 23, 2025 · As laser beams propagate up the density gradient to the critical surface where laser light is efficiently absorbed, they are susceptible to ...
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Plasma wave propagation conditions analysis using the warm multi ...
Plasma wave propagation conditions or accessibility informs whether a wave can propagate to a region, which usually depends on the wave frequency, wave vector, ...
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[PDF] A Short Introduction to Plasma Physics
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Electron density measurements in shock tube using microwave ...
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Study for the microwave interferometer for high densities plasmas on ...
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An incoherent Thomson scattering system for measurements near ...
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What is Thomson Scattering for Plasma Diagnostics
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[PDF] Thomson-Scattering Measurements in the Collective and Non ...
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Design of an O-mode frequency modulated reflectometry system for ...
Reflectometry is a common method for plasma diagnostic, in which microwaves are launched into the plasma and reflected at the critical surfaces.
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[PDF] Refiectometry techniques for density profile measurements on fusion ...
Reflectometry is now widely applied to measure electron density profiles on tokamaks and other toroidal magnetically confined plasmas. This development is ...
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Measurements of density fluctuations in reflectometry of cylindrical ...
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Tungsten Langmuir probes from JET-with the ITER-Like Wall
Dec 8, 2021 · ... density and electron temperature. The density at majority of discharges was in the range from 1 × 1019 m-3 to 1 × 1020 m−3, while the ...
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A new Langmuir probe design to measure plasma properties and ...
This allows us to measure a variety of plasma parameters such as the plasma density, temperature, and space potential in the scrape-off layer (SOL). We will ...
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X-ray Sources Platform | National Ignition Facility & Photon Science
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Development of high intensity X-ray sources at the National Ignition ...
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Long-term comparison of the ionospheric F2 layer electron density ...
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Ionosondes - an overview | ScienceDirect Topics
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A type III radio burst automatic analysis system and statistic results ...
The main defining characteristic of type III radio bursts is that their emission drifts rapidly from high to low frequency, with a drift rate of roughly 100 MHz ...
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[EPUB] Statistics of Low Frequency Cutoffs for Type III Radio Bursts ...
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Grand challenges in low-temperature plasma physics - Frontiers
The temperatures and densities of neutral and charged particles are critical parameters affecting the physical mechanisms within the plasma such as particle ...
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The effects of plasma density and magnetic field on ion temperature ...
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Betatron radiation emitted during the direct laser acceleration of ...
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Single-shot spatiotemporal plasma density measurements with a ...
Sep 12, 2025 · Our work focuses on laser-produced underdense plasmas exhibiting near axial symmetry. In such plasmas, interferometry with a probe pulse is ...