Ghost Lines Research Articles

Physical simulation of the underclad heat affected zone in a reactor pressure vessel to study intergranular cracking

Underclad cracking in nuclear pressure vessels was of significant concern in the 1970s and 1980s before mitigating adjustments were made both to steel compositions and manufacturing practice. Unfortunately, the cracking mechanisms are still not well understood, and this can undermine confidence when changes to cladding operations are under consideration. In this work, Charpy-sized test coupons of 18MND5 steel were subject to a range of thermal cycles that can be experienced by the substrate immediately adjacent to the interface with the overlay. The Charpy test results are considered in combination with analysis of the samples through field emission gun (FEG) scanning electron microscopy (SEM) and wavelength dispersive X-ray spectroscopy (WDXS). The findings suggest that the heat affected zone subject to the coarse grained plus intercritical thermal cycle, before post weld heat treatment, has a peculiar and potentially brittle microstructure. However, in a steel with no obvious macrosegregation, no clear evidence of intergranular cracking in the early stages of the post weld heat treatment could be found. The second part of the work focuses on the effects of segregation regions in large forgings, more specifically ghost lines, on the tendency for underclad cracking. Ghost line regions that subsequently form a coarse grained heat affected zone (CGHAZ), and are then heated to just under the A1 temperature, seem to be the most prone to intergranular cracking. Cracking susceptibility appears to be associated with elevated concentrations of alloying and impurity elements at grain boundaries.

Ghost Line Waves

Time-harmonic electromagnetic plane waves in anisotropic media can exhibit complex-valued wavevectors (with nonzero real and imaginary parts) even in the absence of material dissipation. These peculiar modes, usually referred to as “ghost waves”, hybridize the typical traits of conventional propagating and evanescent waves, displaying both phase accumulation and purely reactive exponential decay away from the direction of the power flow. Their existence has been predicted in several scenarios and has been recently observed experimentally in the form of surface phonon polaritons with complex-valued out-of-plane wavevectors propagating at the interface between air and a natural uniaxial crystal with slanted optical axis. Here, we demonstrate that ghost waves can arise also in lower-dimensional flat-optics scenarios, which are becoming increasingly relevant in the context of metasurfaces and in the field of polaritonics. Specifically, we show that planar junctions between isotropic and anisotropic metasurfaces can support “ghost line waves” that propagate unattenuated along the line interface, exhibiting phase oscillations combined with evanescent decay both in the plane of the metasurface (away from the interface) and out-of-plane in the surrounding medium. Our theoretical results, validated by finite-element numerical simulations, demonstrate a novel form of polaritonic waves with highly confined features, which may provide new opportunities for the control of light at the nanoscale and may find potential applications in a variety of scenarios, including integrated waveguides, nonlinear optics, optical sensing, and subdiffraction imaging.

Effect of Grating Ruling Machine System Errors on Grating Spectral Performance

Grating groove functions scalar theory can describe the light source through the grating interference and diffraction effect of the physical phenomenon of splitting light. It can be used to represent the distribution of the complex amplitude of the illumination light source on the diffraction screen under the action of the diffraction grating and display the information on the diffracted light position and intensity distribution. As a result, the effect of groove error on the intensity of grating ghost lines and stray light is analyzed. This paper presents an energy solution and analysis of the grating periodic groove error utilizing the grating scalar diffraction theory. Based on this, the effect of the core components of the grating ruling machines, such as screw and worm gear, on the stray light and ghost line of the grating is analyzed. Finally, the random groove error is analyzed. The analysis results show that the worm gear error is an important reason for the appearance of ghost lines in the machine ruling grating, and the lead screw error and random grooving error are the main reasons for the stray light produced by the grating. We optimize the worm gear, lead screw, and error compensation system of the grating ruling machine to effectively improve the ruling quality of the grating.

Flexplot: Graphically-based data analysis.

The human visual processing system has enormous bandwidth, able to interpret vast amounts of data in fractions of a second (Otten et al., 2015). Despite this amazing ability, there is a troubling lack of graphics in scientific literature (Healy & Moody, 2014), and the graphics most traditionally used tend to bias perception in unintentional ways (Weissgerber et al., 2015). I suspect the reason for the underuse and misuse of graphics is because sound graphs are difficult to produce with existing software (Wainer, 2010). While ggplot2 allows immense flexibility in creating graphics, its learning curve is quite steep, and even basic graphics require multiple lines of code. flexplot is an R package that aims to address these issues by providing a formula-based suite of tools that simplifies and automates much of the graphical decision-making. Additionally, flexplot pairs well with statistical modeling, making it easy for researchers to produce graphs that map onto statistical procedures. With one-line functions, users can visualize bivariate statistical models (e.g., scatterplots for regression, beeswarm plots for ANOVA/t-tests), multivariate statistical models (e.g., ANCOVA and multiple regression), and even more sophisticated models like multilevel models and logistic regressions. Further, this package utilizes old tools (e.g., added variable plots and coplots) as well as introduces new tools for complex visualizations, including ghost lines and point sampling. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Jerky Motion of the Reaction Front during Discontinuous Dissolution in a Fe-13.5 at.% Zn Alloy.

This paper studies the go- and -stop movement of a receding reaction front (RF) during a discontinuous dissolution (DD) process. A special simulation procedure was applied for the DD reaction to predict a jerky motion of the RF. The Fe-13.5 at.% Zn alloy was selected in which go- and -stop behaviour was revealed in the form of characteristic lines (called “ghost lines”) showing successive positions of receding RF. The results presented for the DD process are quite different from those relevant for the DP reaction at the same Fe-13.5 at.% Zn alloy in terms of go- and -stop motion and movement distance. For the presented case, the go- and -stop periods are relatively long and obtain an order of several dozen seconds, while for the DP reaction, it was only a few seconds. A similar conclusion was formulated after a comparison of the movement distance which, for the DD reaction, is usually longer by 1–2 orders of magnitude. The simulation results of the DD reaction indicate a good agreement with the experimental data presented in the literature for the same dissolution rate. It is necessary to emphasize that the simulation is the only source of data for z parameter changes during the -stop period of the DD reaction.

Discontinuous Dissolution Reaction in a Fe-13.5 at. % Zn Alloy.

The dissolution process of a lamellar structure with α and Γ phases formed during a discontinuous precipitation reaction is investigated here with a Fe-13.5 at. % Zn alloy by means of optical microscopy and scanning and transmission electron microscopy. The α phase is a solute-depleted solid solution and the Γ phase is the intermetallic compound Fe3Zn10. The examination reveals that the dissolution occurs in a discontinuous mode by a receding of the former reaction front of the discontinuous precipitation towards the position of the original grain boundary. A new solid solution in the post-dissolution area is especially inhomogeneous and reflects the former locations of the Γ lamellae (“ghost images”) and the receding reaction front (“ghost lines”). A simulation procedure is applied to determine the Zn concentration profiles left in the post-dissolution region. Their shapes are mostly affected by the Zn content at the positions where the Γ lamellae have just been dissolved, which was also confirmed by the quantitative microchemical analysis.

Identifying the Formation Cause of Ghost Line (Contour) Macrostructure Defects

Identifying the Formation Cause of Ghost Line (Contour) Macrostructure Defects

Ghost Oceans

Ghost Oceans Elizabeth Lee (bio) I stand on a cliff in northern New Mexico, seeing my insides in the rocks. Fascia, red muscle, marbled fat, teeth perched on eroding banks of red and gray earth, dark, ancient blood soaked deep in the sandstone and hard volcanic ash. There's something I know I have to find here if only I could remember what it was. I slide my hands over my tender stomach and the question rises again. Why did the baby die? The white orb of the sun swallows into itself all color and sound and feeling, and my body goes limp. It's my fault. When I first came to New Mexico, I saw all kinds of interesting shapes in the red and purple rocks: Roast beef, ice cream, a camel's back. I felt desperate for a new life. I'd lived for too long in the same 73 degree weather, looking up at a perpetually cloudless sky; I had worked for years in television, anticipating what might look real and feel real, when everything about work and life felt hopelessly fake. I needed a portal to the divine like I needed my childhood wonder again; most importantly, I needed to know there were things in the world that would outlast my body and its memories. After we found out we were pregnant, my husband and I stared at the black computer screen in the doctor's office, studying the ghost lines that curved across each other like tiny white fossils, and waited for a heartbeat. The doctor finally pointed to a white blur. "There it is, and there's the chord." He took a deep breath. "But it's gone into the fallopian tube. I'm sorry, if we leave it there, your tube will rupture. You'll bleed to death." Two days later, he cut out the fallopian tube and the embryo still inside. The blood started immediately, turning my blue-tiled bathroom into a warm vault, thick with the scent of iron and sea salt. I hunched over my abdomen and frowned at all the translucent wires coming out of it. How could I be entirely sure that they hadn't switched out my body parts while I was unconscious and made me part alien? I wiped and saw clumps of dark jelly on the toilet paper. Are you my baby? I asked. Are you? Of course, it was madness, talking to my own blood like that. The baby was gone. My left fallopian tube was gone. But I had lost my mind that day, and it was all I could do not to throw myself out the window when I saw the red home inside my womb had started breaking apart. ________ The wind whips around me. It's like that in New Mexico, coming and going in sudden fits as if to remind the world to stay alert. The sun has made its descent down the sky and the horizon looks like it's on fire, the scorched land more like Mars than Earth. It seems the perfect metaphor for life as my parents talked about it, always describing it as hard and cruel. Survival was the only real true thing, so we'd better prepare. My mother was a quiet and somber woman, letting me twist her coarse hair and drape my leg over her waist while she slept as if she didn't exist at all. I reached for her body at all hours of the day to make [End Page 63] sure she hadn't left me behind. I wasn't sure why, but I always had the sense that she wanted to leave her life, and it frightened me. When her eyes turned to black glass, I knew she had gone deep inside but that she'd be back. I knew great things lived inside that very place because she'd eventually paint beautiful visions on stretched-out silk in her workroom: a woman's face, her collarbone, an ocean wave breaking on sand. She'd eventually hang them up against the tired white walls of our hallways. I don't remember much from that time, but I...

Analysis of the ghost and mirror fields in the Nernst signal induced by superconducting fluctuations

We present a complete analysis of the Nernst signal due to superconducting fluctuations in a large variety of superconductors from conventional to unconventional ones. A closed analytical expression of the fluctuation contribution to the Nernst signal is obtained in a large range of temperature and magnetic field. We apply this expression directly to experimental measurements of the Nernst signal in ${\mathrm{Nb}}_{x}{\mathrm{Si}}_{1\ensuremath{-}x}$ thin films and ${\mathrm{URu}}_{2}{\mathrm{Si}}_{2}$ superconductors. Both magnetic field and temperature dependence of the available data are fitted with very good accuracy using only two fitting parameters, the superconducting temperature ${T}_{\mathrm{c}0}$ and the upper critical field ${H}_{\mathrm{c}2}(0)$. The obtained values agree very well with experimentally obtained values. We also extract the ghost lines (the maximum of the Nernst signal for constant temperature or magnetic field) from the complete expression and also compare it to several experimentally obtained curves. Our approach predicts a linear temperature dependence for the ghost critical field well above ${T}_{\mathrm{c}0}$. Within the errors of the experimental data, this linearity is indeed observed in many superconductors far from ${T}_{\mathrm{c}0}$.

The interaction between vacancies and twin walls, junctions, and kinks, and their mechanical properties in ferroelastic materials

Vacancies strongly interact with twin boundaries and often change dramatically the properties of the ferroelastic material. However, the understanding of this behavior at an atomic-level is still deficient. Here we study vacancy diffusion processes across very large length- and time-scales using a combination of molecular dynamics and Monte Carlo simulations. We find that vacancies reduce their energy by residing at twin boundaries, kinks inside domain boundaries, and junctions between domain boundaries. Vacancies have the largest binding energy inside junctions and co-migrate with the motion of the junctions. For the weaker trapping inside twin boundaries, a “ghost line” may be generated because vacancies do not necessarily diffuse with moving boundaries and are left behind, leaving a trace of a previous position of the domain boundary. Needle twins act as channels for fast diffusion with almost one order of magnitude higher vacancy diffusivity than in bulk. The relative concentration of vacancies at twin boundaries (ρVa) is a function of the average vacancy concentration (CVa) with ρVa ∼ CVaα and α = 0.61, in contrast to that of immobile vacancies case with α = 0.4. The concentration of vacancies at twin boundaries is enriched ca. 5 times at low temperatures. With increasing temperature, the enrichment drops as the trapping potential at the twin boundaries decreases (thermal release). The distribution of energy-drop upon twin pattern evolution follows a power law. The exponent ε increases from ∼1.44 to 2.0 when the vacancy concentration increases. The power law exponent is the same in the athermal region, while Vogel-Fulcher behavior is found at high temperatures.

Spacetime representation of topological phononics

Non-conventional topology of elastic waves arises from breaking symmetry of phononic structures either intrinsically through internal resonances or extrinsically via application of external stimuli. We develop a spacetime representation based on twistor theory of an intrinsic topological elastic structure composed of a harmonic chain attached to a rigid substrate. Elastic waves in this structure obey the Klein–Gordon and Dirac equations and possesses spinorial character. We demonstrate the mapping between straight line trajectories of these elastic waves in spacetime and the twistor complex space. The twistor representation of these Dirac phonons is related to their topological and fermion-like properties. The second topological phononic structure is an extrinsic structure composed of a one-dimensional elastic medium subjected to a moving superlattice. We report an analogy between the elastic behavior of this time-dependent superlattice, the scalar quantum field theory and general relativity of two types of exotic particle excitations, namely temporal Dirac phonons and temporal ghost (tachyonic) phonons. These phonons live on separate sides of a two-dimensional frequency space and are delimited by ghost lines reminiscent of the conventional light cone. Both phonon types exhibit spinorial amplitudes that can be measured by mapping the particle behavior to the band structure of elastic waves.

Rewriting Participation

For his project Everett Ghost Lines, the Japanese author and director Takuya Murakawa writes letters to friends, loose acquaintances, or foreigners, asking them to come to a theatre venue on a designated day at a designated time and perform an action: Sit on a chair and wait for someone who will cut your hair. Stop by the theatre during a run, and lie down on the stage floor next to a duct-taped silhouette. Stand in front of a microphone and tell the audience a story about the sea … The performance starts as scheduled and runs for its full time, whether the letters’ addressees appear or not. The audience gets to read the script, which is being projected on the rear wall of the stage. Written words, thus, fill in for absent performers. Writing assists performance instead of claiming the status of an origin or original inspiration. Thanks to text taking on the role of a temp, an odd hand helping out, Murakawa is able to produce a collective artistic work from the casual, noncommittal, notoriously unreliable relations that make up much of our everyday life. His approach presents an organizational alternative to subjecting art to the sovereign power that is effective in procedures of casting and assures the authority of the legally binding contract, while it also avoids the socio-economic management of networking, which instrumentalizes intimate relations and their emotional intensity in order to generate peer pressure and make ‘free’ cooperation compelling.

Determination of Crystallographic Orientation near a Chill Zone Using Ghost Lines

Many crystals nucleate on the mold surface when the molten alloy is poured in a mold cavity. Because the crystallographic orientations of these crystals are random, the solidified structure near the mold surface is very complex. The ghost lines, which are sometimes thick and the angle between them is not 90 degrees, are often observed in this region. However, if the crystallographic structure of this alloy is cubic, such as bcc or fcc, the ghost lines are very regular. In order to understand the geometry of ghost lines, Al-20 mass%Cu alloys were unidirectionally solidified with constant growth velocity. The solidified structures on the obliquely crossed section were observed. The ghost lines were quite regular and parallel to each other in a solidification grain. The angles and the ratio of the width of ghost lines were measured and crystallographic orientations were estimated using these parameters, based on the solid analytical geometry. EBSD analysis were also performed on the area, where the ghost lines were characterized, and the precise crystallographic orientations were decided. The comparison between both analytical values indicated that the differences between them are within 10 degrees and it can be safely concluded that the estimation for crystallographic orientation using ghost lines agreed well with the EBSD analysis.

RAMAN SCATTERING BY MOLECULAR HYDROGEN AND NITROGEN IN EXOPLANETARY ATMOSPHERES

ABSTRACT An important source of opacity in exoplanet atmospheres at short visible and near-UV wavelengths is Rayleigh scattering of light on molecules. It is accompanied by a related, albeit weaker process—Raman scattering. We analyze the signatures of Raman scattering imprinted in the reflected light and the geometric albedo of exoplanets, which could provide information about atmospheric properties. Raman scattering affects the geometric albedo spectra of planets in the following ways. First, it causes filling-in of strong absorption lines in the incident radiation, thus producing sharp peaks in the albedo. Second, it shifts the wavelengths of spectral features in the reflected light causing the so-called Raman ghost lines. Raman scattering can also cause a broadband reduction of the albedo due to wavelength shifting of a stellar spectrum with red spectral index. Observing the Raman peaks in the albedo could be used to measure the column density of gas, thus providing constraints on the presence of clouds in the atmosphere. Observing the Raman ghost lines could be used to spectroscopically identify the main scatterer in the atmosphere, even molecules like H2 or N2, which do not have prominent spectral signatures in the optical wavelength range. If detected, ghost lines could also provide information about the temperature of the atmosphere. In this paper, we investigate the effects of Raman scattering in hydrogen- and nitrogen-dominated atmospheres. We analyze the feasibility of detecting the signatures of Raman scattering with the existing and future observational facilities, and of using these signatures as probes of exoplanetary atmospheres.

Migration of grain boundaries and triple junctions in high-purity aluminum during annealing after slight cold rolling

Grain orientations and grain boundary migrations near triple junctions in a high purity aluminum were analyzed by electron back scattered diffraction. The results indicate that there are good correlations between the Schmid factors or Taylor factors and the misorientation values of point to original point in grains near the triple junctions in a slightly deformed sample. Grains with higher Schmid factors or lower Taylor factors typically correspond to higher misorientation values near the triple junctions. In a subsequent annealing at 400°C, both grain boundaries and triple junctions migrate, but the former leave ghost lines. During such migration, a grain boundary grows from the grain with lower Schmid factor (higher Taylor factor) into the grain with higher Schmid factor (lower Taylor factor). Usually, the amount of migration of a grain boundary is considerably greater than that of a triple junction, and the grain boundary becomes more curved after migration. These observations indicate that the triple junctions have drag effects on grain boundary migration.

Phenomenology of Hydrogen Flaking in Nuclear Reactor Pressure Vessels*

Abstract Recently, the problem of hydrogen flaking resurfaced, when internal defects were detected in the reactor pressure vessels of two Belgian nuclear power plants. These defects turned out to be hydrogen flakes formed during the fabrication of these pressure vessels. The goal of this publication is to provide important insights into the phenomenon of hydrogen flaking, the different parameters that play a role in the mechanism, as well as the typical morphology and location of these flakes. Therefore an extensive literature study was combined with a detailed metallurgical characterization of a significant number of flakes. Hydrogen flaking is a fabrication problem, which is strongly linked with segregation phenomena. A combination of a sufficient amount of hydrogen, stresses and a sensitive microstructure causes hydrogen flaking. For these reasons hydrogen flakes inside large reactor pressure vessels are found in the so-called ghost lines, which originate from segregation processes during casting.

Ambient betatron motion and its excitation by ``ghost lines'' in Tevatron**Work supported by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy.

Transverse betatron motion of the Tevatron proton beam is measured and analysed. It is shown that the motion is coherent and excited by external sources of unknown origins. Observations of the time varying ``ghost lines'' in the betatron spectra are reported.

DVD Chronicle

DVD Chronicle Jefferson Hunter (bio) Hamlet, directed by Kenneth Branagh (Two-Disc Special Edition, Warner Home Video, 2007) Throne of Blood, directed by Akira Kurosawa (Criterion, 2003) Kwaidan, directed by Masaki Kobayashi (Criterion, 2000) The Devil's Backbone, directed by Guillermo del Toro (Sony, 2004) The Orphanage, directed by Juan Antonio Bayona (Blu-Ray edition, New Line Home Video, 2008) The Haunting, directed by Robert Wise (Warner Home Video, 2003) The Innocents, directed by Jack Clayton (Twentieth-Century Fox, 2005) Cries and Whispers, directed by Ingmar Bergman (Criterion, 2001) If I gave these DVD Chronicles titles, this one would be called "Should We See Them?," the "Them" would refer to ghosts, and the question mark at the end would be real, not rhetorical. To me, it's an unsettled question whether a film's maximum aesthetic and emotional effect is achieved when spirits appear onscreen, or rather when their presence has to be inferred from objects moving inexplicably about, from enigmatic clangings on the sound track, from stares of terror-stricken characters. And indeed if ghosts do appear, what should they look like? Sheeted figures, raw head and bloody bones, ectoplasmic vagueness, the "majesty of buried Denmark" as Hamlet has it? The question was apparently unsettled for Shakespeare, too. His play calls for the ghost of Hamlet's father to be visible to all on the Elsinore ramparts, but then in the boudoir scene with the queen, visible to the prince alone. Failing to see the specter, Gertrude concludes that her son is mad, more forthrightly than any of the critics of the play have ever been able to do. In the filmed versions of Hamlet, this inconsistency in perceiving the ghost has generally been respected. Kenneth Branagh, Franco Zeffirelli, and Rodney Bennett all make the directorial choice to show Hamlet's father on the battlements and in the bedroom, but in the latter scene only to Hamlet; Zeffirelli goes so far as to edit in a p.o.v. shot for Gertrude depicting nothing but vacancy, Shakespeare's "incorporal air." Michael Almereyda's screen version, updated to a world of ruthless CEOs and corporate maneuvering, with a film-within-a-film substituted for Shakespeare's play-within-a-play, also gives Gertrude a p.o.v. shot, but this one shows the ghost, which she then, for unexplained reasons, denies seeing. Perhaps the two most interesting treatments come from more psychologically minded directors, Tony Richardson and Laurence Olivier. In the former's Hamlet, the ghost is never visible to viewers, [End Page 267] except as a bright glare reflected on the prince's face, a sound effect, and a few words in voice-over. Perhaps Richardson's ghost is "really" there, perhaps not, being rather an artifact of Hamlet's perturbed imagination. As for Olivier's ghost, it is visible on the battlements if diaphanously so, out-of-focus, surrounded by clouds of Denham Studio smoke, whispering huskily. By the time the ghost manifests itself in Gertrude's bedroom, however, it has become completely invisible, its photographable form replaced by a beating drum—Hamlet's beating heart, we are no doubt meant to think; the prince almost swoons away. Meanwhile the ghost's lines about the tardy son and the need to whet his almost blunted purpose are rendered in voice-over. Somewhat startlingly, if not indeed comically, Olivier gives the ghost a p.o.v shot, with the camera panning slowly as the unseen figure turns its spectral attention ("look, amazement on thy mother sits") to Gertrude. To argue for the general, not just apparition-depicting, superiority of any one of these screen Hamlets would use up this entire Chronicle, and rather than do that, I will say merely that the version I prefer is Branagh's, largely because of his star performance at the center of the film, but also because of his daringness in overall conception, not to mention his daringness in casting opposite himself actors of such commanding strength as Derek Jacobi (Claudius) and Michael Maloney (Laertes). Maloney, incidentally, appears as a much put-upon actor-director in Branagh's backstage comedy In the Bleak Midwinter, about a production of Hamlet staged in a drafty...

Determination of Orientation of Dendrite from Solidified Structure Observed in 2-D

Solidified structure near the ingot surface is complex and it is difficult to analyze the crystallographic orientation by the solidified structure. The reason of complexity is that the growth direction of dendrite does not agree with the plane for observation. Thus thick lines called ghost lines are often observed. The ghost lines have been characterized in this study.Simplifying the shape of dendrite, the ratio of widths of ghost lines (=γ) has been formulated and the change of γ in orientation of dendrite have been analyzed. It has been found that the angles of ghost lines (η and ξ) as well as γ are the important factors to characterize the solidified structure. In order to determine the spatial orientation of dendrite (θ, φ and β) from η, ξ and γ, the computer program had been developed. Applying this computation method, the solidified structure has been compared with the cross sections of dendrite model. The cross sections obtained by 3D-CAD agreed quite well to the real solidified pattern.

Analysis of Solidified Structure Constituted with Dendrites by Using Solid Analytical Geometry

Solidified structure near the chill zone is complex, since the growth directions of dendrites are not in accordance with the observed plane. Many lines, which are called ghost lines, can be observed in the solidified structure. The ghost lines have been analyzed applying the solid analytical geometry. Dendrite of cubic metals has been simplified with rod as a dendrite trunk and four plates with the same thickness. In order to characterize the solidified pattern, angles of ghost lines (η and ξ) are chosen and calculated using θ, φ and β, which are the important parameters to indicate the dendrite orientation in the space. The calculated results have been confirmed by the experimental results solidified unidirectionally of Al–20mass%Cu alloy. It has been found that the spacing or thickness of ghost lines may be the useful parameters to characterize the spatial array of dendrites with respect to the plane for observation.