Light Scattering Cross Section Research Articles

Constraints on the spin-pole orientation, jet morphology, and rotation of interstellar comet 2I/Borisov with deep HST imaging

ABSTRACT We present high resolution, deep imaging of interstellar comet 2I/Borisov taken with the Hubble Space Telescope/Wide Field Camera 3 (HST/WFC3) on 2019 December 8 UTC and 2020 January 27 UTC (HST GO 16040, PI: Bolin) before and after its perihelion passage in combination with HST/WFC3 images taken on 2019 October 12 UTC and 2019 November 16 UTC (HST GO/DD 16009, PI: Jewitt) before its outburst and fragmentation of 2020 March, thus observing the comet in a relatively undisrupted state. We locate 1–2 arcsec long (2000–3000 km projected length) jet-like structures near the optocentre of 2I that appear to change position angles from epoch to epoch. With the assumption that the jet is located near the rotational pole supported by its stationary appearance on ∼10–100 h time frames in HST images, we determine that 2I’s pole points near α = 322 ± 10° and δ = 37 ± 10° (λ = 341° and β = 48°) and may be in a simple rotation state. Additionally, we find evidence for possible periodicity in the HST time-series light curve on the time-scale of ∼5.3 h with a small amplitude of ∼0.05 mag implying a lower limit on its b/a ratio of ∼1.5 unlike the large ∼2 mag light curve observed for 1I/‘Oumuamua. However, these small light-curve variations may not be the result of the rotation of 2I’s nucleus due to its dust-dominated light-scattering cross-section. Therefore, uniquely constraining the pre-Solar system encounter, pre-outburst rotation state of 2I may not be possible even with the resolution and sensitivity provided by HST observations.

Correction to "Quantification of Material Fluorescence and Light Scattering Cross Sections Using Ratiometric Bandwidth-Varied Polarized Resonance Synchronous Spectroscopy".

Presented herein is the ratiometric bandwidth-varied polarized resonance synchronous spectroscopy (BVPRS2) method for quantification of material optical activity spectra. These include the sample light absorption and scattering cross-section spectrum, the scattering depolarization spectrum, and the fluorescence emission cross-section and depolarization spectrum in the wavelength region where the sample both absorbs and emits. This ratiometric BVPRS2 spectroscopic method is a self-contained technique capable of quantitatively decoupling material fluorescence and light scattering signal contribution to its ratiometric BVPRS2 spectra through the linear curve-fitting of the ratiometric BVPRS2 signal as a function of the wavelength bandwidth used in the PRS2 measurements. Example applications of this new spectroscopic method are demonstrated with materials that can be approximated as pure scatterers, simultaneous photon absorbers/emitters, simultaneous photon absorbers/scatterers, and finally simultaneous phot...

Quantification of Material Fluorescence and Light Scattering Cross Sections Using Ratiometric Bandwidth-Varied Polarized Resonance Synchronous Spectroscopy.

Presented herein is the ratiometric bandwidth-varied polarized resonance synchronous spectroscopy (BVPRS2) method for quantification of material optical activity spectra. These include the sample light absorption and scattering cross-section spectrum, the scattering depolarization spectrum, and the fluorescence emission cross-section and depolarization spectrum in the wavelength region where the sample both absorbs and emits. This ratiometric BVPRS2 spectroscopic method is a self-contained technique capable of quantitatively decoupling material fluorescence and light scattering signal contribution to its ratiometric BVPRS2 spectra through the linear curve-fitting of the ratiometric BVPRS2 signal as a function of the wavelength bandwidth used in the PRS2 measurements. Example applications of this new spectroscopic method are demonstrated with materials that can be approximated as pure scatterers, simultaneous photon absorbers/emitters, simultaneous photon absorbers/scatterers, and finally simultaneous photon absorbers/scatterers/emitters. Because the only instruments needed for this ratiometric BVPRS2 technique are the conventional UV-vis spectrophotometer and spectrofluorometer, this work should open doors for routine decomposition of material UV-vis extinction spectrum into its absorption and scattering component spectra. The methodology and insights provided in this work should be of broad significance to all chemical research that involves photon/matter interactions.

Determining the Liquid Light Scattering Cross Section and Depolarization Spectra Using Polarized Resonance Synchronous Spectroscopy.

Rayleigh scattering is a universal material property because all materials have nonzero polarizability. Reliable quantification of the material light scattering cross section in the liquid phase and its depolarization spectra is, however, challenging due to a host of sample and instrument issues. Using the recently developed polarized resonance synchronous spectroscopic method, we reported the light scattering cross section and depolarization spectra measured for a total of 29 liquids including water, methanol, ethanol, 1-propanol, 1-butanol, dimethylformamide, carbon disulfide, dimethyl sulfoxide, hexane and two hexane isomers (3-methylpentane and 2,3-dimethylbutane), tetrahydrofuran, cyclohexane, acetonitrile, pyridine, chloromethanes including di-, tri, tetrachloromethane, acetone, benzene and eight benzene derivatives (toluene, fluorobenzene, 1,2-, 1,3-, and 1,4-difluorobenzene, chlorobenzene, 1,2- and 1,3-dichlorobenzene, and nitrobenzene). The solvent light scattering depolarization is wavelength-independent for the model solvents, and it varies from 0.023 ± 0.011 for CCl4 to 0.619 ± 0.022 for nitrobenzene. The light scattering cross-section spectra can be approximated with the function of σ(λ) = αλ-4 with the α value varying from 7.2 ± 0.2 × 10-45 cm6 for water to a maximum of 8.5 ± 0.6 × 10-43 cm6 for nitrobenzene. Structural isomerization has no significant effect on either the depolarization or the scattering cross sections for both hexanes and difluorobenzene isomers. This work represents the most comprehensive experimental study on liquid light scattering features. The insight from this work should be important for understanding the correlation between the material structure and optical properties. The described method can be readily implemented by researchers with access to conventional spectrofluorometers equipped with excitation and detection polarizers.

Quantitative and Isolated Measurement of Far-Field Light Scattering by a Single Nanostructure

Optical scattering at the nanoscale continues to lead to physical insights and various applications, including optical antennas and metamaterials and metasurfaces. The differential light-scattering cross section of a $s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}e$ nanostructure, however, has remained elusive, despite being the fundamental parameter for characterizing scattering properties. This article presents quantitative, isolated measurements of the differential cross section and its far-field distribution for a lone gold nanorod. Simulations and modeling are in excellent agreement with the results, showing the phenomena and mechanisms that must be taken into account when engineering devices.

The partial light scattering cross section of spherical particles.

We define the partial scattering cross section and partial efficiencies to demonstrate that the total scattering is the sum of two roughly equal parts: approximately half from the forward scattering lobe due to 2D diffraction by the projected sphere and half from the 2D to 3D diffraction crossover. The first part is at angles such that θ≲λ/D, a result previously known, which can be quite small for large particles. The second part is in a new regime we call the "hump," visible in q-space, which to a good approximation contains the other half of the scattered light. The hump disappears when the imaginary part of the refractive index is significant.

Statistical-thermodynamic model for light scattering from eye lens protein mixtures.

We model light-scattering cross sections of concentrated aqueous mixtures of the bovine eye lens proteins γB- and α-crystallin by adapting a statistical-thermodynamic model of mixtures of spheres with short-range attractions. The model reproduces measured static light scattering cross sections, or Rayleigh ratios, of γB-α mixtures from dilute concentrations where light scattering intensity depends on molecular weights and virial coefficients, to realistically high concentration protein mixtures like those of the lens. The model relates γB-γB and γB-α attraction strengths and the γB-α size ratio to the free energy curvatures that set light scattering efficiency in tandem with protein refractive index increments. The model includes (i) hard-sphere α-α interactions, which create short-range order and transparency at high protein concentrations, (ii) short-range attractive plus hard-core γ-γ interactions, which produce intense light scattering and liquid-liquid phase separation in aqueous γ-crystallin solutions, and (iii) short-range attractive plus hard-core γ-α interactions, which strongly influence highly non-additive light scattering and phase separation in concentrated γ-α mixtures. The model reveals a new lens transparency mechanism, that prominent equilibrium composition fluctuations can be perpendicular to the refractive index gradient. The model reproduces the concave-up dependence of the Rayleigh ratio on α/γ composition at high concentrations, its concave-down nature at intermediate concentrations, non-monotonic dependence of light scattering on γ-α attraction strength, and more intricate, temperature-dependent features. We analytically compute the mixed virial series for light scattering efficiency through third order for the sticky-sphere mixture, and find that the full model represents the available light scattering data at concentrations several times those where the second and third mixed virial contributions fail. The model indicates that increased γ-γ attraction can raise γ-α mixture light scattering far more than it does for solutions of γ-crystallin alone, and can produce marked turbidity tens of degrees celsius above liquid-liquid separation.

Kinetic turbidimetry of patchy colloids aggregation: Latex particles immunoagglutination

The paper is devoted to the problem of the kinetic study of aggregation of particles with several active sites (so called “patchy particles”) using turbidimetry. Patchy particles are known as chemically-anisotropic reactants, and the kinetics theory of their interaction is far from complete. In this work we theoretically derived and experimentally verified analytical expressions, which are convenient for the treatment of the turbidimetric data on the overall optical density change (due to light scattering of the particles) of the aggregating patchy colloid. Light scattering cross sections of the aggregates were calculated using the superposition T-matrix method. Particularly, we used an analytical approximation for the diffusion-limited rate constants of Smoluchowski equations of aggregation of chemically-anisotropic particles (or clusters) with several small active sites assuming that the clusters are of fractal structure. In order to account for the case of a small (<10) number of active sites on particles we used Monte Carlo stochastic algorithm of Gillespie method. For the verification, we carried out experiments on kinetic turbidimetric study of the immunoagglutination of 65nm polystyrene particles covered by anti-CRP IgG antibodies in the water solution of C-reactive protein (CRP). Good agreement of experimental data and theoretical simulations allowed us to evaluate the average number of active antibody molecules per particle as 9.0±2.2 and the affinity of the antigen-antibody complex as 5.5±1.5×107M−1. A significant kinetic effect of the small number of active sites per particle was found.

Rupturing of Biological Spores As a Source of Secondary Particles in Amazonia.

Airborne biological particles, such as fungal spores and pollen, are ubiquitous in the Earth's atmosphere and may influence the atmospheric environment and climate, impacting air quality, cloud formation, and the Earth's radiation budget. The atmospheric transformations of airborne biological spores at elevated relative humidity remain poorly understood and their climatic role is uncertain. Using an environmental scanning electron microscope (ESEM), we observed rupturing of Amazonian fungal spores and subsequent release of submicrometer size fragments after exposure to high humidity. We find that fungal fragments contain elements of inorganic salts (e.g., Na and Cl). They are hygroscopic in nature with a growth factor up to 2.3 at 96% relative humidity, thus they may potentially influence cloud formation. Due to their hygroscopic growth, light scattering cross sections of the fragments are enhanced by up to a factor of 10. Furthermore, rupturing of fungal spores at high humidity may explain the bursting events of new particle formation in Amazonia.

Single and double ultrashort laser pulse scattering by spheroidal metallic nanoparticles

The theory of the ultrashort laser pulse scattered by metallic nanoparticles in the region of surface plasmon resonances is developed in the framework of kinetic approach. For the spheroidal particles, dependence of the light-scattering cross-section on the shape of the particles, the carrier wave frequency, a pulse duration, and other factors is studied. Additionally, an interaction of small metallic particles with double ultrashort pulse is considered. In this case, the energy scattered by the particles demonstrates oscillating behavior when the time delay between pulses changes. Special attention is paid to the contribution of the particle’s surface when the particle’s size is close to the length of free electron path. The difference between the kinetic approach and Mie theory for nonspherical particles has been shown.

UV-Vis Ratiometric Resonance Synchronous Spectroscopy for Determination of Nanoparticle and Molecular Optical Cross Sections.

Demonstrated herein is a UV-vis Ratiometric Resonance Synchronous Spectroscopic (R2S2, pronounced as "R-two-S-two" for simplicity) technique where the R2S2 spectrum is obtained by dividing the resonance synchronous spectrum of a NP-containing solution by the solvent resonance synchronous spectrum. Combined with conventional UV-vis measurements, this R2S2 method enables experimental quantification of the absolute optical cross sections for a wide range of molecular and nanoparticle (NP) materials that range optically from pure photon absorbers or scatterers to simultaneous photon absorbers and scatterers, simultaneous photon absorbers and emitters, and all the way to simultaneous photon absorbers, scatterers, and emitters in the UV-vis wavelength region. Example applications of this R2S2 method were demonstrated for quantifying the Rayleigh scattering cross sections of solvents including water and toluene, absorption and resonance light scattering cross sections for plasmonic gold nanoparticles, and absorption, scattering, and on-resonance fluorescence cross sections for semiconductor quantum dots (Qdots). On-resonance fluorescence quantum yields were quantified for the model molecular fluorophore Eosin Y and fluorescent Qdots CdSe and CdSe/ZnS. The insights and methodology presented in this work should be of broad significance in physical and biological science research that involves photon/matter interactions.

The impact of parameterising light penetration into snow on the photochemical production of NO&amp;lt;sub&amp;gt;&amp;lt;i&amp;gt;x&amp;lt;/i&amp;gt;&amp;lt;/sub&amp;gt; and OH radicals in snow

Abstract. Snow photochemical processes drive production of chemical trace gases in snowpacks, including nitrogen oxides (NOx = NO + NO2) and hydrogen oxide radical (HOx = OH + HO2), which are then released to the lower atmosphere. Coupled atmosphere–snow modelling of theses processes on global scales requires simple parameterisations of actinic flux in snow to reduce computational cost. The disagreement between a physical radiative-transfer (RT) method and a parameterisation based upon the e-folding depth of actinic flux in snow is evaluated. In particular, the photolysis of the nitrate anion (NO3-), the nitrite anion (NO2-) and hydrogen peroxide (H2O2) in snow and nitrogen dioxide (NO2) in the snowpack interstitial air are considered. The emission flux from the snowpack is estimated as the product of the depth-integrated photolysis rate coefficient, v, and the concentration of photolysis precursors in the snow. The depth-integrated photolysis rate coefficient is calculated (a) explicitly with an RT model (TUV), vTUV, and (b) with a simple parameterisation based on e-folding depth, vze. The metric for the evaluation is based upon the deviation of the ratio of the depth-integrated photolysis rate coefficient determined by the two methods, vTUV/vze, from unity. The ratio depends primarily on the position of the peak in the photolysis action spectrum of chemical species, solar zenith angle and physical properties of the snowpack, i.e. strong dependence on the light-scattering cross section and the mass ratio of light-absorbing impurity (i.e. black carbon and HULIS) with a weak dependence on density. For the photolysis of NO2, the NO2- anion, the NO3- anion and H2O2 the ratio vTUV/vze varies within the range of 0.82–1.35, 0.88–1.28, 0.93–1.27 and 0.91–1.28 respectively. The e-folding depth parameterisation underestimates for small solar zenith angles and overestimates at solar zenith angles around 60° compared to the RT method. A simple algorithm has been developed to improve the parameterisation which reduces the ratio vTUV/vze to 0.97–1.02, 0.99–1.02, 0.99–1.03 and 0.98–1.06 for photolysis of NO2, the NO2- anion, the NO3- anion and H2O2 respectively. The e-folding depth parameterisation may give acceptable results for the photolysis of the NO3- anion and H2O2 in cold polar snow with large solar zenith angles, but it can be improved by a correction based on solar zenith angle and for cloudy skies.

Prediction of inelastic light scattering spectra from electronic collective excitations in GaAs/AlGaAs core-multishell nanowires

We predict inelastic light scattering spectra from electron collective excitations in a coaxial quantum well embedded in a core-multishell $\mathrm{GaAs}/\mathrm{AlGaAs}$ nanowire. The complex composition, the hexagonal cross section, and the remote doping of typical samples are explicitly included, and the free electron gas is obtained by a density functional theory (DFT) approach. Inelastic light scattering cross sections due to charge and spin collective excitations belonging to quasi-one-dimensional (1D) and quasi-2D states, which coexist in such radial heterostructures, are predicted in the nonresonant approximation from a fully three-dimensional multisubband time-dependent DFT (TDDFT) formalism. We show that collective excitations can be classified in azimuthal, radial, and longitudinal excitations, according to the associated density fluctuations, and we suggest that their character can be exposed by specific spectral dispersion of inelastic light scattering along different planes of the heterostructure.

The effects of additional black carbon on the albedo of Arctic sea ice: variation with sea ice type and snow cover

Abstract. The response of the albedo of bare sea ice and snow-covered sea ice to the addition of black carbon is calculated. Visible light absorption and light-scattering cross-sections are derived for a typical first-year and multi-year sea ice with both "dry" and "wet" snow types. The cross-sections are derived using data from a 1970s field study that recorded both reflectivity and light penetration in Arctic sea ice and snow overlying sea ice. The variation of absorption cross-section over the visible wavelengths suggests black carbon is the dominating light-absorbing impurity. The response of first-year and multi-year sea ice albedo to increasing black carbon, from 1 to 1024 ng g−1, in a top 5 cm layer of a 155 cm-thick sea ice was calculated using a radiative-transfer model. The albedo of the first-year sea ice is more sensitive to additional loadings of black carbon than the multi-year sea ice. An addition of 8 ng g−1 of black carbon causes a decrease to 98.7% of the original albedo for first-year sea ice compared to a decrease to 99.7% for the albedo of multi-year sea ice, at a wavelength of 500 nm. The albedo of sea ice is surprisingly unresponsive to additional black carbon up to 100 ng g−1 . Snow layers on sea ice may mitigate the effects of black carbon in sea ice. Wet and dry snow layers of 0.5, 1, 2, 5 and 10 cm depth were added onto the sea ice surface. The albedo of the snow surface was calculated whilst the black carbon in the underlying sea ice was increased. A layer of snow 0.5 cm thick greatly diminishes the effect of black carbon in sea ice on the surface albedo. The albedo of a 2–5 cm snow layer (less than the e-folding depth of snow) is still influenced by the underlying sea ice, but the effect of additional black carbon in the sea ice is masked.

Theory of scattering of electromagnetic waves of the microwave range in a turbid medium

The coefficient of extinction of electromagnetic waves of the microwave range due to their scattering from clusters suspended in an amorphous medium and responsible for turbidity is calculated. Turbidity resembles the case when butter clusters transform water into milk. In the case under investigation, the clusters are conductors (metallic or semiconducting). The extinction coefficient is connected in a familiar way with the cross section of light scattering from an individual cluster. A new formula is derived for the light scattering cross section in the case when damping of oscillations of an electron is due only to spontaneous emission of light quanta. In this case, the resonant scattering cross section for light can be very large. It is shown that this can be observed only in a whisker nanocluster. In addition, the phonon energy on a whisker segment must be higher than the photon energy, which is close to the spacing between the electron energy levels in the cluster.

Sticky-Sphere Mixture Model for Light Scattering from Concentrated Eye Lens Protein Mixtures

We use a sticky-sphere mixture liquid structure model to analyze static light scattering efficiency data obtained from aqueous mixtures of gamma and alpha crystallin eye lens proteins. The model incorporates (i) hard-sphere interactions between alpha crystallins, leading to short-range ordering and transparency at high concentration, (ii) short-range attractive interactions between gamma crystallins, which can produce intense light scattering and liquid-liquid phase separation, and (iii) both excluded-volume interactions and short-range attractions between alpha and gamma crystallins, which also sensitively affect transparency [Dorsaz:2009,Dorsaz:2011]. We derive general expressions for the scattered light intensity within the model, and compare the model with data from aqueous gamma/alpha mixtures. We show that the sticky-sphere mixture model can provide accurate representations of the observed static light scattering cross sections, including their concave-down dependence on alpha/gammaB composition at intermediate concentrations, combined with the concave up and more complicated, temperature-sensitive dependence on relative protein composition at high, more physiological protein concentrations [Thurston:2006]. We use the model to help study possible implications for molecular mechanisms of cataract, by changing gamma-gamma, gamma-alpha, and alpha-alpha interactions. The sticky-sphere mixture model exhibits non-monotonic dependence of light scattering intensity and thermodynamic stability on gamma-alpha attraction strength, consistent with simulation and perturbation theory [Dorsaz:2011]. The model predicts that gamma-alpha mixture light scattering is quite sensitive to gamma-alpha interactions, consistent with previous analysis of a congenital cataract [Banerjee:2011]. The sticky-sphere mixture model is useful for quantitatively analyzing light scattering of concentrated aqueous crystallin mixtures, relevant for understanding lens cytoplasm.

Optical Bragg, atomic Bragg and cavity QED detections of quantum phases and excitation spectra of ultracold atoms in bipartite and frustrated optical lattices

Ultracold atoms loaded on optical lattices can provide unprecedented experimental systems for the quantum simulations and manipulations of many quantum phases and quantum phase transitions between these phases. However, so far, how to detect these quantum phases and phase transitions effectively remains an outstanding challenge. In this paper, we will develop a systematic and unified theory of using the optical Bragg scattering, atomic Bragg scattering or cavity QED to detect the ground state and the excitation spectrum of many quantum phases of interacting bosons loaded in bipartite and frustrated optical lattices. The physically measurable quantities of the three experiments are the light scattering cross sections, the atom scattered clouds and the cavity leaking photons respectively. We show that the two photon Raman transition processes in the three detection methods not only couple to the density order parameter, but also the valence bond order parameter due to the hopping of the bosons on the lattice. This valence bond order coupling is very sensitive to any superfluid order or any valence bond (VB) order in the quantum phases to be probed. These quantum phases include not only the well-known superfluid and Mott insulating phases, but also other important phases such as various kinds of charge density waves (CDW), valence bond solids (VBS), and CDW-VBS phases with both CDW and VBS orders unique to frustrated lattices, and also various kinds of supersolids. We analyze respectively the experimental conditions of the three detection methods to probe these various quantum phases and their corresponding excitation spectra. We also address the effects of a finite temperature and a harmonic trap. We contrast the three scattering methods with recent in situ measurements inside a harmonic trap and argue that the two kinds of measurements are complementary to each other. The combination of both kinds of detection methods could be used to match the combination of Scanning tunneling microscopy (STM), the Angle Resolved Photo Emission spectroscopy (ARPES) and neutron scattering in condensed matter systems, therefore achieve the putative goals of quantum simulations

Biosensing potential of three-layered gold–dielectric–gold nanoshells: sensitivity of interdistance of resonance light scattering peaks to the local dielectric environment

The resonance light scattering cross section of the three-layered gold–dielectric–gold nanoshell is calculated using the quasi-static theory. The results show that three-layered gold–dielectric–gold nanoshells have three types of resonance light scattering spectra under different geometrical parameters, which are explained using plasmon hybridization theory. In the common type containing two resonance light scattering peaks, the longer wavelength peak red-shifts while the shorter one blue-shifts as the refractive index of the physical environment increase. This physical environment dependent interdistance enhancing between the two peaks shows the potential of the three-layered gold–dielectric–gold nanoshells used as biosensor based on resonance light scattering. The followed calculations about the comparison of sensitivity to the local dielectric environment between the three- and two-layered gold nanoshells further verify that point. At last, the calculated results that the three-layered gold–dielectric–gold nanoshells with small inner core and thin outer gold shell have the highest sensitivity are also obtained.

Optical properties of polystyrene black dyed microspheres

The light absorption cross section is one of the most important optical properties along with the light scattering and the light extinction cross sections. The light scattering cross section when added to the light absorption cross section produces the light extinction cross section. Light absorption cross sections of polystyrene black dyed microspheres were measured by a multi angle absorption photometer (MAAP) together with the combination of a differential mobility analyzer (DMA) and a condensation particle counter (CPC). The MAAP is a filter based instrument that can measure light absorption properties without artifacts by using backward light scattering intensity measurements to compensate. Preliminary result for the 980 nm sized polystyrene black dyed microspheres depicts that the absorption cross section is about 1.5 × 10−14 m2 per sphere. In addition, the light absorption cross section of 233 nm and 10 μm spheres were measured. The near field scattering image data are shown for three polystyrene black dyed microspheres. The implication of the optical properties for the different sized black spheres is that the absorption coefficient and scattering image data depend on the size of the spheres.

Light scattering in inhomogeneous Tomonaga-Luttinger liquids

We derive the dynamical structure factor for an inhomogeneous Tomonaga-Luttinger liquid as can be formed in a confined strongly interacting one-dimensional gas. In view of current experimental progress in the field, we provide a simple analytic expression for the light-scattering cross section, requiring only the knowledge of the density dependence of the ground-state energy, as they can be extracted e.g. from exact or Quantum Monte Carlo techniques, and a Thomas-Fermi description. We apply the result to the case of one-dimensional quantum bosonic gases with dipolar interaction in a harmonic trap, using an energy functional deduced from Quantum Monte Carlo computations. We find an universal scaling behavior peculiar of the Tomonaga-Luttinger liquid, a signature that can be eventually probed by Bragg spectroscopy in experimental realizations of such systems.