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At least 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Effects of load proportioning on the capacity of multiple-hole composite joints

This study addresses the issue of adjusting the proportion of load transmitted by each hole in a multiple-hole joint so that the joint capacity is a maximum. Specifically two-hole-in-series joints are examined. The results indicate that when each hole reacts 50% of the total load, the joint capacity is not a maximum. One hole generally is understressed at joint failure. The algorithm developed to determine the load proportion at each hole which results in maximum capacity is discussed. The algorithm includes two-dimensional finite-element stress analysis and failure criteria. The algorithm is used to study the effects of joint width, hole spacing, and hole to joint-end distance on load proportioning and capacity. To study hole size effects, two hole diameters are considered. Three laminates are considered: a quasi-isotropic laminate; a cross-ply laminate; and a 45 degree angle-ply laminate. By proportioning the load, capacity can be increased generally from 5 to 10%. In some cases a greater increase is possible.

Hyer, M. W.↗
NASA NTRS1993

Primordial black holes in globular clusters

It has recently been recognized that significant numbers of medium-mass back holes (of order 10 solar masses) should form in globular clusters during the early stages of their evolution. Here we explore the dynamical and observational consequences of the presence of such a primordial black-hole population in a globular cluster. The holes initially segregate to the cluster cores, where they form binary and multiple black-hole systems. The subsequent dynamical evolution of the black-hole population ejects most of the holes on a relatively short timescale: a typical cluster will retain between zero and four black holes in its core, and possibly a few black holes in its halo. The presence of binary, triple, and quadruple black-hole systems in cluster cores will disrupt main-sequence and giant stellar binaries; this may account for the observed anomalies in the distribution of binaries in globular clusters. Furthermore, tidal interactions between a multiple black-hole system and a red giant star can remove much of the red giant's stellar envelope, which may explain the puzzling absence of larger red giants in the cores of some very dense clusters.

Sigurdsson, Steinn↗
NASA NTRS1995

Heating of coronal holes by the resonant absorption and dissipation of Alfven waves and its relation to solar wind acceleration

Coronal hole regions are well known sources of high-speed solar wind, however to account for the observed properties of the solar wind a source of momentum and heat must be included. Alfven waves were suggested as the possible source of heating that accelerates the solar wind. We investigate the propagation of the Alfven waves in coronal holes via numerical solution of the linearized 2-D resistive MHD equations in slab geometry. The Alfven waves are driven at the lower boundary of the coronal hole and propagate into the corona. The waves are reflected at the coronal hole boundary and part of the wave energy leaks out of the coronal hole. We compare the calculated wavelengths and the attenuation rate of the fast mode Alfven waves in the leaky waveguide formed by the coronal hole with the analytical ideal MHD solutions. The formation of resonance heating layers is found to occur when shear Alfven waves propagate in an inhomogeneous coronal hole. The heating is enhanced when fast mode waves couple to the shear Alfven waves. The narrow heating layers are formed near the location of the ideal resonance, which might occur near the coronal hole boundary for a nearly constant density coronal hole, surrounded by a higher density plasma. We investigate the dependence of the heating on the driver frequency, the Lundquist number, and on the heliocentric distance. and find that the low frequency Alfven waves can be an efficient source of heating at large distances from the Sun. We discuss the relation of our results to the observed properties of high-speed solar wind and coronal holes.

Ofman, L.↗
NASA NTRS2003

High-Time-Resolution Study of Magnetic Holes in the Solar Wind

The objectives of this investigation are to determine the internal plasma structure of kinetic-scale and larger scale magnetic holes, and to determine their stability, their source mechanism(s), and their spatial extent. It is also of importance to determine the relationship between kinetic-scale holes and long-duration holes. As smaller and smaller magnetic depressions are investigated in order to make this a complete study, a robust criterion is necessary for distinguishing magnetic holes from random or unresolvable fluctuations in the interplanetary magnetic field. In order to resolve this ambiguity, we obtained from the MFI experiments magnetic field measurements from the WIND spacecraft at a time resolution of 46 to 184 ms over certain periods. We have also devised a measure of certainty for magnetic hole detections. The certainty factor, q, is defined as the difference between the mean magnetic field in the hole and the local magnetic field, in units of the local standard deviation of the field strength. For fullest generality, it is necessary to calculate this q over the range of available scales of interest, from 60 ms up to 300 s. This technique establishes a two dimensional matrix of relative probabilities that a hole of some duration (d) might exist in the data set at a given time (t). In identifying q-peaks in time and duration, we also come upon a natural method for distinguishing holes with internal structure from multiple holes in close proximity or holes nested inside of others. If two q-peaks are more than a half-width apart, they are simply said to be separate events.

Lazarus, Alan↗
NASA NTRS2005

Method for Estimating Thread Strength Reduction of Damaged Parent Holes with Inserts

During normal assembly and disassembly of bolted-joint components, thread damage and/or deformation may occur. If threads are overloaded, thread damage/deformation can also be anticipated. Typical inspection techniques (e.g. using GO-NO GO gages) may not provide adequate visibility of the extent of thread damage. More detailed inspection techniques have provided actual pitch-diameter profiles of damaged-hardware holes. A method to predict the reduction in thread shear-out capacity of damaged threaded holes has been developed. This method was based on testing and analytical modeling. Test samples were machined to simulate damaged holes in the hardware of interest. Test samples containing pristine parent-holes were also manufactured from the same bar-stock material to provide baseline results for comparison purposes. After the particular parent-hole thread profile was machined into each sample a helical insert was installed into the threaded hole. These samples were tested in a specially designed fixture to determine the maximum load required to shear out the parent threads. It was determined from the pristine-hole samples that, for the specific material tested, each individual thread could resist an average load of 3980 pounds. The shear-out loads of the holes having modified pitch diameters were compared to the ultimate loads of the specimens with pristine holes. An equivalent number of missing helical coil threads was then determined based on the ratio of shear-out loads for each thread configuration. These data were compared with the results from a finite element model (FEM). The model gave insights into the ability of the thread loads to redistribute for both pristine and simulated damage configurations. In this case, it was determined that the overall potential reduction in thread load-carrying capability in the hardware of interest was equal to having up to three fewer threads in the hole that bolt threads could engage. One- half of this potential reduction was due to local pitch-diameter variations and the other half was due to overall pitch-diameter enlargement beyond Class 2 fit. This result was important in that the thread shear capacity for this particular hardware design was the limiting structural capability. The details of the method development, including the supporting testing, data reduction and analytical model results comparison will be discussed hereafter.

Johnson, David L.↗
NASA NTRS2018

Generation of Electron Whistler Waves at the Mirror Mode Magnetic Holes: MMS Observations and PIC Simulation

The Magnetospheric Multiscale mission has observed electron whistler waves at the center and at the edges of magnetic holes in the dayside magnetosheath. The magnetic holes are nonlinear mirror structures since their magnitude is anticorrelated with particle density. In this article, we examine the growth mechanisms of these whistler waves and their interaction with the host magnetic hole. In the observations, as magnetic holes develop and get deeper, an electron population gets trapped and develops a temperature anisotropy favorable for whistler waves to be generated. In addition, the decrease in magnetic field magnitude and the increase in density reduce the electron resonance energy, which promotes the electron cyclotron resonance. To investigate this process, we used expanding box particle-in-cell simulations to produce the mirror instability, which then evolve into magnetic holes. The simulation shows that whistler waves can be generated at the center and edges of magnetic holes, which reproduces the primary features of the MMS observations. The simulation shows that the electron temperature anisotropy develops in the center of the magnetic hole once the mirror instability reaches its nonlinear stage of evolution. The plasma is then unstable to whistler waves at the minimum of the magnetic field structures. In the saturation regime of mirror instability, when magnetic holes are developed, the electron temperature anisotropy appears at the edges of the holes and electron distributions become more isotropic at the magnetic field minimum. At the edges, the expansion of magnetic holes decelerates the electrons, which leads to temperature anisotropies.

Magnetospheric Multiscale↗
DOE OSTI2014

Geometrical and band-structure effects on phonon-limited hole mobility in rectangular cross-sectional germanium nanowires

We calculated the phonon-limited hole mobility in rectangular cross-sectional [001], [110], [111], and [112]-oriented germanium nanowires, and the hole transport characteristics were investigated. A tight-binding approximation was used for holes, and phonons were described by a valence force field model. Then, scattering probability of holes by phonons was calculated taking account of hole-phonon interaction atomistically, and the linearized Boltzmann's transport equation was solved to calculate the hole mobility at low longitudinal field. The dependence of the hole mobility on nanowire geometry was analyzed in terms of the valence band structure of germanium nanowires, and it was found that the dependence was qualitatively reproduced by considering an average effective mass and the density of states of holes. The calculation revealed that [110] germanium nanowires with large height along the [001] direction show high hole mobility. Germanium nanowires with this geometry are also expected to exhibit high electron mobility in our previous work, and thus they are promising for complementary metal-oxide-semiconductor (CMOS) applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

How the Universe postpones the evaporation and curtails the quantum spreading of black holes

Black holes are expected to evaporate through the process of Hawking radiation. This process is expected to cause the uncertainty in a black hole’s position to grow to ~ $M^4/M^3_{pl}$ over the course of it’s lifetime, even as its momentum spreads only by ~ $M_{Pl}$ For the black holes that have been observed, which have M ≥ M ⊙ this greatly exceeds the Hubble volume. However, the decay of black holes and their quantum spreading, are delayed in the Universe while the influx of energy into the black holes exceeds their Hawking luminosity. We show that for these M ≥ M ⊙ black holes, their decay outside galaxies and clusters is prevented far longer than it takes the black holes to be dragged well beyond the Hubble horizon, where their eventual decay occurs away from the prying eyes of any observer who has not hitched a ride with them. Meanwhile, black holes in an observer’s galaxy or cluster are themselves prevented from decaying long past the extinction of the last stars, and at least until their galaxy/cluster is swept clean of dark matter, in $\gg$ 10 25 y. Even then, if the black holes become unbound, then they are dragged beyond the Hubble radius before undergoing significant decay; if not, they remain in bound orbits, spreading at most over a subvolume of the galaxy/cluster, and long localized by scatterings to much smaller volumes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

From rotating to charged black holes and back again

The mild form of the Weak Gravity Conjecture (WGC) requires higher derivative corrections to extremal charged black holes to increase their charge-to-mass ratio. This allows decay via emission of a smaller extremal black hole. In this paper, we investigate if similar constraints hold for extremal rotating black holes. We do so by considering the leading higher derivative corrections to the four-dimensional Kerr black hole and five-dimensional Myers-Perry black hole. We use a known mapping of these rotating solutions to a four-dimensional non-rotating dyonic Kaluza-Klein black hole and impose the WGC on this charged solution. Going back again to the rotating solutions, this fixes the sign of the corrections to the rotating extremality bounds. The sign of the corrections is non-universal, depending on the black hole under consideration. We argue that this is not at odds with black hole decay, because of the presence of a superradiant instability that persists in the extremal limit. When this instability is present, the WGC is implied for the four-dimensional charged black hole.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2012

Drift hole structure and dynamics with turbulence driven flows

The role of turbulence driven flows in describing drift hole structure and dynamics is discussed. Turbulence driven flows enter the plasma medium response and alter drift hole structures by changing the screening length of the drift hole potential. Specifically, turbulence driven flows shift the drift hole potential radially, and absorb drift hole energy via the hole-flow resonance. It is shown that the absorption shifts the phase of a momentum flux, and so enables the irreversible coupling of drift holes to turbulence driven flows. We show that drift holes and turbulence driven flows are dynamically coupled, and self-regulate each other, so that a stationary state can be achieved with non-zero turbulence driven flows. As an application, a bound on the fluctuation amplitude in the coupled system is derived. The bound is obtained by requiring that the resultant zonal flow velocity should be smaller than the critical flow velocity for the drift hole potential to be self-bound (i.e., the velocity that the screening length be positive). The result predicts Double-Vertical-Line {phi}(tilde sign) Double-Vertical-Line {sub max}{sup 2}{approx}({nu}{sub d}/{omega}{sub ci})(k{sub Parallel-To }/k{sub y}), where zonal flow damping appears as a control parameter. The implications of this result for the problem of edge-core coupling (i.e., explaining turbulence and transport in 'No Man's Land') are discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

A hole accelerator for InGaN/GaN light-emitting diodes

The quantum efficiency of InGaN/GaN light-emitting diodes (LEDs) has been significantly limited by the insufficient hole injection, and this is caused by the inefficient p-type doping and the low hole mobility. The low hole mobility makes the holes less energetic, which hinders the hole injection into the multiple quantum wells (MQWs) especially when a p-type AlGaN electron blocking layer (EBL) is adopted. In this work, we report a hole accelerator to accelerate the holes so that the holes can obtain adequate kinetic energy, travel across the p-type EBL, and then enter the MQWs more efficiently and smoothly. In addition to the numerical study, the effectiveness of the hole accelerator is experimentally shown through achieving improved optical output power and reduced efficiency droop for the proposed InGaN/GaN LED.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2016

Ionization equilibrium at the transition from valence-band to acceptor-band migration of holes in boron-doped diamond

A quasi-classical model of ionization equilibrium in the p-type diamond between hydrogen-like acceptors (boron atoms which substitute carbon atoms in the crystal lattice) and holes in the valence band (v-band) is proposed. The model is applicable on the insulator side of the insulator–metal concentration phase transition (Mott transition) in p-Dia:B crystals. The densities of the spatial distributions of impurity atoms (acceptors and donors) and of holes in the crystal are considered to be Poissonian, and the fluctuations of their electrostatic potential energy are considered to be Gaussian. The model accounts for the decrease in thermal ionization energy of boron atoms with increasing concentration, as well as for electrostatic fluctuations due to the Coulomb interaction limited to two nearest point charges (impurity ions and holes). The mobility edge of holes in the v-band is assumed to be equal to the sum of the threshold energy for diffusion percolation and the exchange energy of the holes. On the basis of the virial theorem, the temperature T{sub j} is determined, in the vicinity of which the dc band-like conductivity of holes in the v-band is approximately equal to the hopping conductivity of holes via the boron atoms. For compensation ratio (hydrogen-like donor to acceptor concentration ratio) K ≈ 0.15 and temperature T{sub j}, the concentration of “free” holes in the v-band and their jumping (turbulent) drift mobility are calculated. Dependence of the differential energy of thermal ionization of boron atoms (at the temperature 3T{sub j}/2) as a function of their concentration N is calculated. The estimates of the extrapolated into the temperature region close to T{sub j} hopping drift mobility of holes hopping from the boron atoms in the charge states (0) to the boron atoms in the charge states (−1) are given. Calculations based on the model show good agreement with electrical conductivity and Hall effect measurements for p-type diamond with boron atom concentrations in the range from 3 × 10{sup 17} to 3 × 10{sup 20 }cm{sup −3}, i.e., up to the Mott transition. The model uses no fitting parameters.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2011

ON THE HIGH-FREQUENCY QUASI-PERIODIC OSCILLATIONS FROM BLACK HOLES

We apply the global mode analysis, which has been recently developed for the modeling of kHz quasi-periodic oscillations (QPOs) from neutron stars, to the inner region of an accretion disk around a rotating black hole. Within a pseudo-Newtonian approach that keeps the ratio of the radial epicyclic frequency {kappa} to the orbital frequency {Omega} the same as the corresponding ratio for a Kerr black hole, we determine the innermost disk region where the hydrodynamic modes grow in amplitude. We find that the radiation flux emerging from the inner disk has the highest values within the same region. Using the flux-weighted averages of the frequency bands over this region we identify the growing modes with highest frequency branches {Omega} + {kappa} and {Omega} to be the plausible candidates for the high-frequency QPO pairs observed in black hole systems. The observed frequency ratio around 1.5 can therefore be understood naturally in terms of the global free oscillations in the innermost region of a viscous accretion disk around a black hole without invoking a particular resonance to produce black hole QPOs. Although the frequency ratio ({Omega} + {kappa})/({Omega}) is found to be not sensitive to the black hole's spin which is good for explaining the high-frequency QPOs, it may work as a limited diagnostic of the spin parameter to distinguish black holes with very large spin from the slowly rotating ones. Within our model we estimate the frequency ratio of a high-frequency QPO pair to be greater than 1.5 if the black hole is a slow rotator. For fast rotating black holes, we expect the same ratio to be less than 1.5.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

New observational constraints on the growth of the first supermassive black holes

We constrain the total accreted mass density in supermassive black holes at z > 6, inferred via the upper limit derived from the integrated X-ray emission from a sample of photometrically selected galaxy candidates. Studying galaxies obtained from the deepest Hubble Space Telescope images combined with the Chandra 4 Ms observations of the Chandra Deep Field-South, we achieve the most restrictive constraints on total black hole growth in the early universe. We estimate an accreted mass density <1000 M {sub ☉} Mpc{sup –3} at z ∼ 6, significantly lower than the previous predictions from some existing models of early black hole growth and earlier prior observations. These results place interesting constraints on early black hole growth and mass assembly by accretion and imply one or more of the following: (1) only a fraction of the luminous galaxies at this epoch contain active black holes; (2) most black hole growth at early epochs happens in dusty and/or less massive—as yet undetected—host galaxies; (3) there is a significant fraction of low-z interlopers in the galaxy sample; (4) early black hole growth is radiatively inefficient, heavily obscured, and/or due to black hole mergers as opposed to accretion; or (5) the bulk of the black hole growth occurs at late times. All of these possibilities have important implications for our understanding of high-redshift seed formation models.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

The black hole in the compact, high-dispersion galaxy NGC 1271

Located in the Perseus cluster, NGC 1271 is an early-type galaxy with a small effective radius of 2.2 kpc and a large bulge stellar velocity dispersion of 276 km s{sup −1} for its K-band luminosity of 8.9×10{sup 10} L{sub ⊙}. We present a mass measurement for the black hole in this compact, high-dispersion galaxy using observations from the Near-infrared Integral Field Spectrometer on the Gemini North telescope assisted by laser guide star adaptive optics, large-scale integral field unit observations with PPAK at the Calar Alto Observatory, and Hubble Space Telescope WFC3 imaging observations. We are able to map out the stellar kinematics both on small spatial scales, within the black hole sphere of influence, and on large scales that extend out to four times the galaxy’s effective radius. We find that the galaxy is rapidly rotating and exhibits a sharp rise in the velocity dispersion. Through the use of orbit-based stellar dynamical models, we determine that the black hole has a mass of (3.0{sub −1.1}{sup +1.0})×10{sup 9} M{sub ⊙} and the H-band stellar mass-to-light ratio is 1.40{sub −0.11}{sup +0.13} Υ{sub ⊙} (1σ uncertainties). NGC 1271 occupies the sparsely populated upper end of the black hole mass distribution but is very different from the brightest cluster galaxies (BCGs) and giant elliptical galaxies that are expected to host the most massive black holes. Interestingly, the black hole mass is an order of magnitude larger than expectations based on the galaxy’s bulge luminosity but is consistent with the mass predicted using the galaxy’s bulge stellar velocity dispersion. More compact, high-dispersion galaxies need to be studied using high spatial resolution observations to securely determine black hole masses, as there could be systematic differences in the black hole scaling relations between these types of galaxies and the BCGs/giant ellipticals, thereby implying different pathways for black hole and galaxy growth.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

A CONNECTION BETWEEN PLASMA CONDITIONS NEAR BLACK HOLE EVENT HORIZONS AND OUTFLOW PROPERTIES

Accreting black holes are responsible for producing the fastest, most powerful outflows of matter in the universe. The formation process of powerful jets close to black holes is poorly understood, and the conditions leading to jet formation are currently hotly debated. In this paper, we report an unambiguous empirical correlation between the properties of the plasma close to the black hole and the particle acceleration properties within jets launched from the central regions of accreting stellar-mass and supermassive black holes. In these sources the emission of the plasma near the black hole is characterized by a power law at X-ray energies during times when the jets are produced. We find that the photon index of this power law, which gives information on the underlying particle distribution, correlates with the characteristic break frequency in the jet spectrum, which is dependent on magnetohydrodynamical processes in the outflow. The observed range in break frequencies varies by five orders of magnitude in sources that span nine orders of magnitude in black hole mass, revealing a similarity of jet properties over a large range of black hole masses powering these jets. This correlation demonstrates that the internal properties of the jet rely most critically on the conditions of the plasma close to the black hole, rather than other parameters such as the black hole mass or spin, and will provide a benchmark that should be reproduced by the jet formation models.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Black holes and the multiverse

Vacuum bubbles may nucleate and expand during the inflationary epoch in the early universe. After inflation ends, the bubbles quickly dissipate their kinetic energy; they come to rest with respect to the Hubble flow and eventually form black holes. The fate of the bubble itself depends on the resulting black hole mass. If the mass is smaller than a certain critical value, the bubble collapses to a singularity. Otherwise, the bubble interior inflates, forming a baby universe, which is connected to the exterior FRW region by a wormhole. A similar black hole formation mechanism operates for spherical domain walls nucleating during inflation. As an illustrative example, we studied the black hole mass spectrum in the domain wall scenario, assuming that domain walls interact with matter only gravitationally. Our results indicate that, depending on the model parameters, black holes produced in this scenario can have significant astrophysical effects and can even serve as dark matter or as seeds for supermassive black holes. The mechanism of black hole formation described in this paper is very generic and has important implications for the global structure of the universe. Baby universes inside super-critical black holes inflate eternally and nucleate bubbles of all vacua allowed by the underlying particle physics. The resulting multiverse has a very non-trivial spacetime structure, with a multitude of eternally inflating regions connected by wormholes. If a black hole population with the predicted mass spectrum is discovered, it could be regarded as evidence for inflation and for the existence of a multiverse.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2011

Schwarzschild black holes as unipolar inductors: Expected electromagnetic power of a merger

The motion of a Schwarzschild black hole with velocity v{sub 0}={beta}{sub 0}c through a constant magnetic field B{sub 0} in vacuum induces a component of the electric field along the magnetic field, generating a nonzero second Poincare electromagnetic invariant {sup *}F{center_dot}F{ne}0. This will produce (e.g., via radiative effects and vacuum breakdown) an electric charge density of the order of {rho}{sub ind}=B{sub 0{beta}0}/(2{pi}eR{sub G}), where R{sub G}=2GM/c{sup 2} is the Schwarzschild radius and M is the mass of the black hole; the charge density {rho}{sub ind} is similar to the Goldreich-Julian density. The magnetospheres of moving black holes resemble in many respects the magnetospheres of rotationally-powered pulsars, with pair formation fronts and outer gaps, where the sign of the induced charge changes. As a result, the black hole will generate bipolar electromagnetic jets each consisting of two counter-aligned current flows (four current flows total), each carrying an electric current of the order I{approx_equal}eB{sub 0}R{sub G{beta}0}. The electromagnetic power of the jets is L{approx_equal}(GM){sup 2}B{sub 0}{sup 2{beta}}{sub 0}{sup 2}/c{sup 3}; for a particular case of merging black holes the resulting Poynting power is L{approx_equal}(GM){sup 3}B{sub 0}{sup 2}/(c{sup 5}R), where R is the radius of the orbit. In addition, in limited regions near the horizon the first electromagnetic invariant changes sign, so that the induced electric field becomes larger than the magnetic field, E>B. As a result, there will be local dissipation of the magnetic field close to the horizon, within a region with the radial extent {Delta}R{approx_equal}R{sub G{beta}0}. The total energy loss from a system of merging black holes is a sum of two components with similar powers, one due to the rotation of space-time within the orbit, driven by the nonzero angular momentum in the system, and the other due to the linear motion of the black holes through the magnetic field. Since the resulting electrodynamics is in many respects similar to pulsars, merging black holes may generate coherent radio and high energy emission beamed approximately along the orbital normal. In addition, merging black holes may produce observable wind-driven cavities.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗