Quasi-Isoscattering Stealthy Hyperuniform Materials

• Quasi-isoscattering stealthy hyperuniform materials are disordered systems engineered to eliminate low-k scattering while maintaining a nearly flat scattering profile beyond the stealth regime.
• Design methodologies, such as Fourier-space simulated annealing and collective-coordinate optimization, precisely tune reciprocal-space parameters to create isotropic transparent band gaps.
• These materials pave the way for innovative applications in wave control, energy-saving devices, and photonic or acoustic systems by leveraging controlled spectral density constraints.

Quasi-isoscattering stealthy hyperuniform materials are disordered hyperuniform media whose reciprocal-space correlations simultaneously enforce a stealth condition—vanishing scattering for a finite neighborhood of the origin, typically $S(\mathbf{k})=0$ or $\tilde\chi_V(\mathbf{k})=0$ for $0<|\mathbf{k}|<K$—and a nearly angle-independent or band-flat scattering response outside that region over a prescribed finite interval or angular sector. In the two-phase setting, they are heterogeneous materials with suppressed large-scale volume-fraction fluctuations, isotropic statistics, and no Bragg peaks; in point-pattern settings, they are particle arrangements with vanishing low-$k$ density fluctuations. Across these formulations, the central design idea is to sculpt the structure factor or spectral density so that long-wavelength scattering channels are eliminated while the residual scattering intensity is nearly constant over a target band, yielding transparency, isotropic band gaps, and tunable transport or effective-medium behavior (Chen et al., 2017, Leseur et al., 2015, Alhaïtz et al., 2023).

1. Definitions and reciprocal-space descriptors

For point scatterers at positions $\{\mathbf r_i\}$, the normalized structure factor is

$$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$

A stealthy hyperuniform distribution is one for which

$$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$

with $S(\mathbf 0)=N$ in the normalization used for the ultrasonic experiments. In this formulation, stealthy hyperuniformity suppresses all density fluctuations at wavenumbers below the stealth radius $K$ (Alhaïtz et al., 2023).

For two-phase media, the relevant descriptor is the spectral density. If $I(\mathbf x)$ is the indicator function of phase 1 and $\tilde\chi_V(\mathbf{k})=0$0 its volume fraction, then the two-point autocovariance is

$\tilde\chi_V(\mathbf{k})=0$1

and the spectral density is its Fourier transform,

$\tilde\chi_V(\mathbf{k})=0$2

On a digitized $\tilde\chi_V(\mathbf{k})=0$3 periodic grid, Chen and Torquato wrote

$\tilde\chi_V(\mathbf{k})=0$4

where $\tilde\chi_V(\mathbf{k})=0$5. Hyperuniformity enforces $\tilde\chi_V(\mathbf{k})=0$6 through the sum rule $\tilde\chi_V(\mathbf{k})=0$7, and stealthy hyperuniformity further requires $\tilde\chi_V(\mathbf{k})=0$8 for $\tilde\chi_V(\mathbf{k})=0$9 (Chen et al., 2017).

In this literature, “quasi-isoscattering” refers to a nearly flat scattering response once the stealth region is exited. Two complementary operational definitions appear. One is a prescribed piecewise-constant target spectrum $0<|\mathbf{k}|<K$0 with $0<|\mathbf{k}|<K$1 for $0<|\mathbf{k}|<K$2 and $0<|\mathbf{k}|<K$3 for $0<|\mathbf{k}|<K$4, producing a flat scattering plateau over $0<|\mathbf{k}|<K$5 (Chen et al., 2017). The other is an angular criterion in which the measured transmission $0<|\mathbf{k}|<K$6 has negligible variance with respect to $0<|\mathbf{k}|<K$7, quantified by

$0<|\mathbf{k}|<K$8

with quasi-isoscattering corresponding to $0<|\mathbf{k}|<K$9 over the target bands (Alhaïtz et al., 2023).

2. Statistical-mechanical basis and diffraction phenomenology

The ensemble theory of stealthy hyperuniform disordered ground states gives a reciprocal-space basis for quasi-isoscattering. For isotropic pair potentials with Fourier transform $k$0 supported on $k$1, any configuration that forces all collective-density modes to zero inside $k$2 is a ground state. In any ensemble of such ground states,

$k$3

and more generally

$k$4

Torquato and coauthors then introduced a pseudo–hard–sphere ansatz in Fourier space, mapping $k$5 to the pair statistics of equilibrium hard spheres with an effective packing fraction proportional to the stealthiness parameter $k$6. For sufficiently small $k$7, the consequence is that x-ray or neutron intensity $k$8 shows suppression of small-angle scattering together with a nearly flat large-angle response, $k$9, aside from a small $\{\mathbf r_i\}$0 ripple that decays to unity by $\{\mathbf r_i\}$1. The resulting diffraction pattern is a one-ring pattern of near-uniform intensity, which they explicitly described as quasi-isoscattering (Torquato et al., 2015).

A complementary route begins from the effective single-scattering cross section in a correlated medium,

$\{\mathbf r_i\}$2

For stealth hyperuniform point patterns, if $\{\mathbf r_i\}$3 inside a small-$\{\mathbf r_i\}$4 region and approximately unity outside, the angular dependence of the differential cross section is inherited directly from the low-$\{\mathbf r_i\}$5 suppression. In the perturbative regime analyzed for dense hyperuniform materials, the transparent regime is obtained when $\{\mathbf r_i\}$6 and $\{\mathbf r_i\}$7, while a quasi-isoscattering regime can be engineered when $\{\mathbf r_i\}$8: scattering vanishes in a narrow forward cone defined by $\{\mathbf r_i\}$9, and for $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$0 the product $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$1 becomes essentially constant if $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$2 is nearly angle-independent, as in small dipoles in 2D TE polarization (Leseur et al., 2015).

This combination of exact stealth constraints and weakly structured post-cutoff intensity explains why stealthy hyperuniform media occupy an intermediate regime between crystals and generic disordered media. They inherit isotropy and the absence of Bragg peaks from disorder, but they inherit sharp reciprocal-space exclusion zones and associated wave-control functionality from reciprocal-space order. A plausible implication is that quasi-isoscattering is not an ancillary feature but the natural continuation of the stealth constraint once the first allowed scattering shell is engineered to be weakly structured.

3. Construction and inverse-design methodologies

The most explicit two-phase construction protocol is the Fourier-space simulated-annealing method of Chen and Torquato. One seeks a binary pattern $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$3 whose spectral density matches a prescribed target $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$4 by minimizing

$$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$5

The implementation initializes a random pattern at fixed $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$6, swaps one pixel of phase 1 with one of phase 2 at each Monte Carlo step, updates $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$7 and $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$8 in $$S(\mathbf q)=\frac{1}{N}\sum_{i,j=1}^N e^{\,\mathrm{i}\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} =\frac{1}{N}\left|\sum_{i=1}^N e^{\,\mathrm{i}\mathbf q\cdot\mathbf r_i}\right|^2.$$9 time using the incremental formulas, accepts or rejects moves by the Metropolis rule at temperature $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$0, and terminates when $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$1, for example $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$2. Quasi-isoscattering is introduced directly in reciprocal space through a piecewise-constant radial target: $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$3 Because only the radial average is constrained, the resulting material is statistically isotropic; phase-inversion symmetry may also be enforced to simplify the parameter space and, in bicontinuous settings, to support optimized transport (Chen et al., 2017).

Point-pattern constructions often use collective-coordinate optimization. In the 2D ultrasonic rod experiments, the arrangements were generated by an iterative minimization that drives $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$4 for $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$5 while preserving the exclusion $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$6. In the mid-infrared semiconductor implementation, the target was $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$7 for $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$8 and $$S(\mathbf q)=0 \quad \text{for all } |\mathbf q|<K,$$9 otherwise, minimized over a periodic supercell until $S(\mathbf 0)=N$0 for all $S(\mathbf 0)=N$1 (Alhaïtz et al., 2023, Gallego et al., 2024).

A more recent phase-sensitive design route casts a two-phase material as a heterogeneous network. The local optical potential is decomposed as $S(\mathbf 0)=N$2, and the weighted graph edge between particles $S(\mathbf 0)=N$3 and $S(\mathbf 0)=N$4 is

$S(\mathbf 0)=N$5

The graph is then decomposed into $S(\mathbf 0)=N$6-$S(\mathbf 0)=N$7, $S(\mathbf 0)=N$8-$S(\mathbf 0)=N$9, and $K$0-$K$1 subnetworks. An insertion algorithm places each new particle at the position minimizing

$K$2

thereby steering intra-phase and inter-phase correlations separately while almost preserving the overall scattering response in the target stealth region (Youn et al., 30 Jul 2025).

Multispecies generalizations replace a single structure factor by multihyperuniform statistics across $K$3 species. Keeney and coauthors used hard-core exclusion for all species together with same-species soft-shell repulsions, alternating growth and soft-shell relaxation until the final packing fraction $K$4–$K$5. The computed observables include species-resolved and total $K$6, $K$7, total $K$8, number variance $K$9, and nearest-neighbor distributions. In that framework, the resulting $I(\mathbf x)$0 can be nearly flat and zero over $I(\mathbf x)$1, producing what they called high-“bandwidth” isotropic suppression, or quasi-isoscattering stealthiness (Keeney et al., 7 Oct 2025).

4. Transparency, attenuation, and effective-medium behavior

In both point-pattern and two-phase formulations, the elimination of low-$I(\mathbf x)$2 modes produces transparency. For two-phase media in the long-wavelength limit, the effective complex dielectric constant $I(\mathbf x)$3 has an imaginary part proportional to

$I(\mathbf x)$4

Because a hyperuniform medium satisfies $I(\mathbf x)$5, Chen and Torquato argued that $I(\mathbf x)$6 in lowest order. Stealthy hyperuniformity extends this transparency to all $I(\mathbf x)$7, since no Fourier channels exist to scatter into for $I(\mathbf x)$8, leading to transmission approximately equal to unity in that regime (Chen et al., 2017).

In one-dimensional layered dielectrics, the nonlocal strong-contrast theory gives an explicit route from spectral density to transmittance. The effective transverse dielectric constant is expressed in terms of the attenuation function

$I(\mathbf x)$9

so that $\tilde\chi_V(\mathbf{k})=0$00. If the medium is truly stealthy hyperuniform, $\tilde\chi_V(\mathbf{k})=0$01 for $\tilde\chi_V(\mathbf{k})=0$02, then $\tilde\chi_V(\mathbf{k})=0$03 for

$\tilde\chi_V(\mathbf{k})=0$04

yielding a perfect transparency interval and implying no Anderson localization in the infinite-size limit. By contrast, non-stealthy hyperuniform media do not possess a truly transparent interval, because $\tilde\chi_V(\mathbf{k})=0$05 for all $\tilde\chi_V(\mathbf{k})=0$06 (Kim et al., 2023).

The same theory also clarifies how quasi-isoscattering may be programmed: because attenuation is controlled by $\tilde\chi_V(\mathbf{k})=0$07, a nearly constant target transmittance $\tilde\chi_V(\mathbf{k})=0$08 over a finite band can be approximated by prescribing $\tilde\chi_V(\mathbf{k})=0$09 over the doubled band. The flat-top spectral-density construction given for layered media is the one-dimensional analogue of the piecewise-constant $\tilde\chi_V(\mathbf{k})=0$10 used in two-phase Fourier-space design (Kim et al., 2023, Chen et al., 2017).

For dense point scatterers, the perturbative transparency criterion adds a transport condition. When $\tilde\chi_V(\mathbf{k})=0$11, the leading-order terms in the phase function cancel, and the first non-vanishing contribution to $\tilde\chi_V(\mathbf{k})=0$12 scales like $\tilde\chi_V(\mathbf{k})=0$13. The resulting criterion

$\tilde\chi_V(\mathbf{k})=0$14

shows that a slab can remain transparent at densities for which an uncorrelated disordered material would be opaque due to multiple scattering (Leseur et al., 2015).

5. Experimental realizations and isotropic band-gap behavior

The most direct experimental evidence for quasi-isoscattering in stealth hyperuniform media has been obtained with ultrasound in 2D arrangements of steel rods immersed in water. Two complementary setups were used: a plane-wave slab and a point-source disk. In the slab experiment, a 500 kHz-central ultrasonic transducer launched bursts at normal incidence onto a rectangular stealth hyperuniform slab, and the transmitted field was recorded to determine the complex transmission $\tilde\chi_V(\mathbf{k})=0$15, the effective attenuation

$\tilde\chi_V(\mathbf{k})=0$16

and the scattering mean-free path $\tilde\chi_V(\mathbf{k})=0$17. In the $\tilde\chi_V(\mathbf{k})=0$18, $\tilde\chi_V(\mathbf{k})=0$19 slab, $\tilde\chi_V(\mathbf{k})=0$20 up to $\tilde\chi_V(\mathbf{k})=0$21 MHz and $\tilde\chi_V(\mathbf{k})=0$22, while a complete band gap appeared near $\tilde\chi_V(\mathbf{k})=0$23 MHz. In the point-source geometry, the directivity pattern $\tilde\chi_V(\mathbf{k})=0$24 demonstrated that $\tilde\chi_V(\mathbf{k})=0$25 over both the transparency band and the full gap for $\tilde\chi_V(\mathbf{k})=0$26 and $\tilde\chi_V(\mathbf{k})=0$27, whereas random and quasi-crystalline patterns showed $\tilde\chi_V(\mathbf{k})=0$28 (Alhaïtz et al., 2023).

These experiments support a dual interpretation of quasi-isoscattering. Below the cut-off frequency $\tilde\chi_V(\mathbf{k})=0$29, long-range spectral constraints suppress all accessible scattering channels and produce omnidirectional transparency. Near the first Bragg frequency $\tilde\chi_V(\mathbf{k})=0$30, short- and medium-range real-space correlations encoded in the oscillatory $\tilde\chi_V(\mathbf{k})=0$31 generate a complete, isotropic band gap. The coexistence of isotropic transparency and isotropic band-gap formation is a defining feature of the stealth hyperuniform architecture in wave physics (Alhaïtz et al., 2023).

A photonic realization in the mid-infrared used a hole-based stealthy hyperuniform pattern fabricated on a quantum cascade layer substrate. The design used a $\tilde\chi_V(\mathbf{k})=0$32 periodic supercell with $\tilde\chi_V(\mathbf{k})=0$33 holes, average spacing $\tilde\chi_V(\mathbf{k})=0$34, hole radius $\tilde\chi_V(\mathbf{k})=0$35, and $\tilde\chi_V(\mathbf{k})=0$36, corresponding to $\tilde\chi_V(\mathbf{k})=0$37. The measured photonic band gap appeared around $\tilde\chi_V(\mathbf{k})=0$38 with a gap-midgap ratio of $\tilde\chi_V(\mathbf{k})=0$39, and the reflection spectrum was unchanged for all in-plane rotational angle measurements. At $\tilde\chi_V(\mathbf{k})=0$40, the reflectance variation remained within $\tilde\chi_V(\mathbf{k})=0$41 dB as $\tilde\chi_V(\mathbf{k})=0$42 swept from $\tilde\chi_V(\mathbf{k})=0$43 to $\tilde\chi_V(\mathbf{k})=0$44, and the in-plane variance satisfied $\tilde\chi_V(\mathbf{k})=0$45 across the band gap, compared with $\tilde\chi_V(\mathbf{k})=0$46 for a triangular photonic crystal (Gallego et al., 2024).

Taken together, the ultrasonic and mid-infrared experiments show that quasi-isoscattering is not restricted to weak single-scattering regimes. It persists in dense multiple-scattering media, in complete band-gap settings, and in metamaterial implementations where angular isotropy is probed directly by rotation or point-source measurements.

6. Phase topology, multiphase generalizations, applications, and limitations

In two-phase composites, reciprocal-space control couples directly to topology and transport. Chen and Torquato constructed a family of phase-inversion-symmetric materials with variable topological connectedness that achieves a well-known explicit formula for the effective electrical or thermal conductivity, and they also designed a disordered stealthy hyperuniform dispersion with nearly optimal effective conductivity while remaining statistically isotropic. Their framework emphasizes that quasi-isoscattering design need not be restricted to wave attenuation: the same spectral-density control can be used to tune connectivity, bicontinuity, and effective transport (Chen et al., 2017).

Phase-sensitive engineering extends this principle by allowing distinct phases to acquire different local environments without substantially altering the low-$\tilde\chi_V(\mathbf{k})=0$47 scattering signature. In the heterogeneous-network formulation, varying $\tilde\chi_V(\mathbf{k})=0$48 changes the relative weight of bipartite and unipartite links. For $\tilde\chi_V(\mathbf{k})=0$49, both phases exhibit the textbook stealthy hyperuniform $\tilde\chi_V(\mathbf{k})=0$50 decay of scaled number variance; for $\tilde\chi_V(\mathbf{k})=0$51, phase $\tilde\chi_V(\mathbf{k})=0$52 remains stealthy hyperuniform-like while phase $\tilde\chi_V(\mathbf{k})=0$53 becomes nearly Poisson-like; and for $\tilde\chi_V(\mathbf{k})=0$54, phase $\tilde\chi_V(\mathbf{k})=0$55 develops a pronounced clustering peak at intermediate $\tilde\chi_V(\mathbf{k})=0$56, even though all designs maintain

$\tilde\chi_V(\mathbf{k})=0$57

in the normalized units used there (Youn et al., 30 Jul 2025).

Multihyperuniform particle composites widen the design space further. Balanced species number ratios produce the strongest hyperuniform signatures; examples in the summary include $\tilde\chi_V(\mathbf{k})=0$58 for a binary $\tilde\chi_V(\mathbf{k})=0$59 system at $\tilde\chi_V(\mathbf{k})=0$60, and $\tilde\chi_V(\mathbf{k})=0$61 within numerical precision for $\tilde\chi_V(\mathbf{k})=0$62 in a four-species system at $\tilde\chi_V(\mathbf{k})=0$63. Within the Born approximation, $\tilde\chi_V(\mathbf{k})=0$64, so isotropic suppression of $\tilde\chi_V(\mathbf{k})=0$65 over $\tilde\chi_V(\mathbf{k})=0$66 yields an angle-independent scattering dip. Reported applications include isotropic structural coloration, enhanced absorption, stealth or transparent coatings, and engineered dielectric properties that facilitate transmission while suppressing scattering (Keeney et al., 7 Oct 2025).

The principal control parameters recur across formulations. In two-phase pixel or voxel designs, $\tilde\chi_V(\mathbf{k})=0$67 and $\tilde\chi_V(\mathbf{k})=0$68 fix the stealth gap and the width of the quasi-isoscattering band; $\tilde\chi_V(\mathbf{k})=0$69 controls connectivity; and $\tilde\chi_V(\mathbf{k})=0$70 with pixel size $\tilde\chi_V(\mathbf{k})=0$71 sets the $\tilde\chi_V(\mathbf{k})=0$72-space resolution, with $\tilde\chi_V(\mathbf{k})=0$73 and $\tilde\chi_V(\mathbf{k})=0$74. Practical guidance includes choosing $\tilde\chi_V(\mathbf{k})=0$75, using annealing schedules $\tilde\chi_V(\mathbf{k})=0$76 with $\tilde\chi_V(\mathbf{k})=0$77–$\tilde\chi_V(\mathbf{k})=0$78, and terminating when $\tilde\chi_V(\mathbf{k})=0$79 (Chen et al., 2017). In point-pattern realizations, the stealthiness degree $\tilde\chi_V(\mathbf{k})=0$80 parametrizes the number of constrained modes and, in the rod experiments, obeys

$\tilde\chi_V(\mathbf{k})=0$81

In that convention, $\tilde\chi_V(\mathbf{k})=0$82 corresponds to a random pattern and $\tilde\chi_V(\mathbf{k})=0$83 tends to a perfect crystal (Alhaïtz et al., 2023). The multihyperuniform framework, however, states that increasing $\tilde\chi_V(\mathbf{k})=0$84 narrows the stealthy band $\tilde\chi_V(\mathbf{k})=0$85 and can drive ordering. This suggests that reported bandwidth trends are formulation-dependent and should be interpreted within the specific construction protocol (Keeney et al., 7 Oct 2025).

Fabrication routes include 3D printing, lithographic etching, direct hole-patterning in semiconductor platforms, and layer-by-layer construction for one-dimensional media. Reported and suggested applications include energy-saving materials, batteries, aerospace uses, wave-guide, filter, cloaking, transparent low-haze composites, non-iridescent color coatings, and low-loss metamaterial slabs (Chen et al., 2017, Alhaïtz et al., 2023, Keeney et al., 7 Oct 2025). The main limitations are also explicit in the literature: strict stealth constraints over a finite $\tilde\chi_V(\mathbf{k})=0$86-band are challenging to realize experimentally; finite-size effects require domain sizes exceeding approximately $\tilde\chi_V(\mathbf{k})=0$87 by an order of magnitude in some designs; and several analytical descriptions, including the pseudo–hard–sphere ansatz, are controlled most clearly for sufficiently small $\tilde\chi_V(\mathbf{k})=0$88, with reported validity up to roughly $\tilde\chi_V(\mathbf{k})=0$89 in $\tilde\chi_V(\mathbf{k})=0$90 (Keeney et al., 7 Oct 2025, Torquato et al., 2015).

Quasi-isoscattering stealthy hyperuniform materials therefore constitute a reciprocal-space design class rather than a single material family. Their unifying property is the deliberate co-design of a low-$\tilde\chi_V(\mathbf{k})=0$91 exclusion zone and an isotropic, weakly structured post-cutoff response. In the available formulations, this combination produces transparent intervals, complete isotropic band gaps, phase-sensitive microstructures with nearly preserved overall scattering, and a route from two-point statistics to wave functionality across acoustic, electromagnetic, and multiphase composite settings.