Lattice Unit Research Articles

Multiscale design optimization of lattice structures considering dynamic response minimization and reaction force uniformity

A multiscale design optimization model is proposed for a vibrating hierarchical lattice structure, considering minimization of vibration response and uniformity of reaction forces at the fixed boundary. Two different lattice unit cells are hybridized to improve the manufacturability and mechanical properties of the lattice structures. Lattice microstructures are designed and controlled by multiple design variables. Surrogate models are used to fit the equivalent performance with the design variables. An optimization formulation is established for minimizing the displacement response amplitude at specified locations of the structure. The variance and mean value of the reaction force are introduced as constraints for the uniform reaction force at the structure boundary. Sensitivities of the objective and constraint functions are derived with the adjoint method. Finally, the effectiveness of the method is verified by several numerical examples, and the influence of the variance constraint of the reaction force at the structure boundary is discussed.

Coherent control of tunneling-based photonic lattice unit cells through an induced chiral atomic medium

Coherent control of tunneling-based photonic lattice unit cells through an induced chiral atomic medium

Design and Evaluation of 3D-Printed Lattice Structures as High Flow Rate Aerosol Filters.

Aerosol contamination presents significant challenges across various industries, ranging from healthcare to manufacturing. Over the past few years, open foam filters have gained prominence for their ability to efficiently capture particles while allowing reasonable airflow. In this work, we present the use of 3D-printed idealized open foam-like lattice structures as aerosol filtration media, leveraging advances in additive manufacturing to generate these highly tunable and modular filters. Using parametric design approaches, we fabricated lattice filters with four different unit cell geometries (Cubic, Kelvin, Octahedron, and Weaire-Phelan) via Digital Light Synthesis 3D printing and characterized these structures with X-ray microcomputed tomography. We compared the aerosol filtration performance of the different lattice unit cell geometries using 1 μm polystyrene latex (PSL) aerosol particles, finding the filtration performance to be positively correlated with the single-unit-cell specific surface area. We then expanded our evaluation of deposition efficiency in Kelvin cell lattice structures of varied porosities, again finding a correlation between the specific surface area and deposition performance. Experimental analysis confirmed that deposition primarily occurs through impaction and electrostatic mechanisms within the parameter space. Overall, our findings demonstrate that unit-cell-based lattices can achieve a wide range of aerosol filtration efficiencies (∼10-100%) across various operating conditions (1-4 m/s superficial velocity), offering a highly tunable in-line filtration medium capable of maintaining high efficiency even at elevated airflow rates. This work not only provides essential guidelines for designing and manufacturing 3D-printed lattices as customizable aerosol filters but also highlights the current limitations and challenges in producing these structures.

Unveiling the pressure-induced physical variations in Cs2SnBr6 and Cs2SnCl6: A DFT study

This study employs the FP-LAPW method based on DFT to investigate the structural, electronic, and optical properties of Cs2SnBr6 and Cs2SnCl6 under a range of pressures up to 30[Formula: see text]GPa. We use the TB-mBJ functional to analyze the electronic and optical properties of the materials, while the GGA–PBEsol is employed to study their structural properties. Applying pressure causes a decrease in the unit cell volume and lattice constant of the double perovskites. At zero pressure, both Cs2SnBr6 and Cs2SnCl6 exhibit a direct bandgap. The bandgaps of both the materials decrease as the pressure increases from 0[Formula: see text]GPa to 20[Formula: see text]GPa but slightly increase for the increase in pressure from 20[Formula: see text]GPa to 30[Formula: see text]GPa. The DOS analysis conducted at 0[Formula: see text]GPa and 30[Formula: see text]GPa demonstrates a decrease in the number of states per[Formula: see text]eV as the pressure increases, along with an increase in bandwidth and a shift of the peaks toward negative energy. The electron density contours exhibit the presence of both covalent and ionic bonds at zero pressure, whereas at a pressure of 30[Formula: see text]GPa, the covalent bond strength is enhanced. Optical properties of Cs2SnBr6 and Cs2SnCl6 were investigated over a pressure range of 0–30[Formula: see text]GPa and a photon energy range of 0–10[Formula: see text]eV. With increasing pressure, the initial peak in these parameters was red-shifted due to a decrease in bandgap energy, while the remaining peaks blue-shifted due to the broadening of conduction and valence bands.

Exploring the Impact of Elevated Pressure on the Structural Dynamics of TlInS2 Crystal (I): A First-Principles Investigation with XRD, Raman, and Hirshfeld Topology

The structural dynamics of Thallium Indium Disulfide (TlInS2) crystal under elevated pressures were comprehensively investigated using a combination of X-ray diffraction (XRD), Raman spectroscopy via first-principles calculations, and Hirshfeld surface (HS) analysis. This study aims to elucidate pressure-induced modifications in the crystal structure and their associated properties of TlInS2 crystal. The findings reveal significant structural transformations as pressure increases, with XRD patterns indicating lattice compression and a critical transition from semiconductor to conductor at pressure P=6.25GPa [42]. The calculations predict a reduction in bond lengths, specifically S1-In1 and S1-Tl2, alongside a corresponding decrease in lattice parameters and unit cell volume. Raman spectroscopy corroborates these changes through shifts in phonon modes. The HS analysis provides further insights into pressure-dependent alterations in intermolecular interactions, revealing changes in voids and surface characteristics. The data gained from TlInS2's structural behavior under pressure, contributing to the optimization of its functional properties for pressure-tunable applications in high-performance electronic and optoelectronic devices.

Sharp volume and multiplicity bounds for Fano simplices

We present sharp upper bounds on the volume, Mahler volume and multiplicity for Fano simplices depending on the dimension and Gorenstein index. These bounds rely on the interplay between lattice simplices and unit fraction partitions. Moreover, we present an efficient procedure for explicitly classifying Fano simplices of any dimension and Gorenstein index, and we carry out the classification up to dimension four for various Gorenstein indices.

Tuning band gap and improving optoelectronic properties of lead-free halide perovskites FrMI3 (M = Ge, Sn) under hydrostatic pressure

In the field of optoelectronic applications, inorganic cubic halide perovskites have turned into a leading choice for commercialization. The physical properties of cubic FrMI3 (M = Ge, Sn) halide perovskites under pressure are explored at multiple pressures up to 10 GPa using ab initio Density-Functional Theory due to their significant importance. The structural accuracy, reflected in lattice parameters and unit cell volume, closely corresponds to previously reported data. FrGeI3 and FrSnI3 exhibited direct electronic band gaps of 0.65 eV and 0.46 eV with GGA-PBE functional, respectively, indicating their semiconducting nature at 0 GPa. The enhanced band gaps obtained employing the HSE06 potential are 1.61 eV and 1.28 eV for the corresponding perovskites. FrGeI3 and FrSnI3 shift from semiconducting to metallic states under pressures of 4.5 GPa and 3 GPa, respectively, thereby enhancing the conductivity. Pressure also improves optical functions, indicating both perovskites may be employed in high-efficiency optoelectronic devices that operate within the visible and ultra-violet (UV) range. At 10 GPa pressure, both FrGeI3 and FrSnI3 demonstrate the sharpest absorption spikes in the UV region, approximately at 2.61 × 105 cm−1 and 2.42 × 105 cm−1, respectively, resulting in sharp peaks in optical conductivity of approximately 5.98 1/fs for FrGeI3 and 5.49 1/fs for FrSnI3. FrGeI3 consistently outperforms FrSnI3, offering superior electronic and optical characteristics. Additionally, pressure modifies the mechanical features of entitled perovskites while conserving stability. The anisotropic and ductile properties become more pronounced under pressure. The outcomes of this study are expected to propel the advancement of lead-free optoelectronic devices, driving innovation in sustainable energy solutions.

Mesoscale modeling on the influence of surfactants on seepage law during water injection in coal

To investigate the mesoscopic influence of surfactants on seepage law during water injection in coal seam, this paper innovatively establishes a fluid transport lattice Boltzmann (LBM) model by incorporating the seepage resistance generated from the porous media and external forces, which embodies the impact of wettability degree resulted from cocamidopropyl betaine (CAB), sodium dodecyl sulfate (SDS), and coconutt diethanol amide (CDEA) reagents at a 0.1% concentration. The main conclusions derived from this investigation are as follows: Firstly, as the lattice number in the X direction increases, the average seepage velocities in coal samples treated by deionized water, 0.1% CAB, 0.1% SDS, and 0.1% CDEA reagents (Nos. 1, 2, 3, and 4) exhibit three distinct stages: rapid decline, slow decline, and steady decline; in comparison to raw coal sample, modified coal samples demonstrate decreases of 20.84%, 33.91%, and 61.70%, respectively. Secondly, the critical values of displacement pressure difference exist during the phenomenon that modified reagents spread out in the entire flow channel, which are 3.5, 3.5, and 5.2 MPa, respectively, for coal samples Nos. 2, 3, and 4; this signifies that surpassing these critical values help prevent issues such as blank belts within the wetting range and insufficient dust control. Finally, at a displacement pressure difference of 0.01 (lattice unit), the average velocity ratios for samples (Nos. 2, 3, and 4) are 0.78, 0.56, and 0.37 (lattice unit), respectively; notably, the water flow velocities in modified coal samples are lower compared to that in raw coal sample, indicating that the addition of surfactants impede the seepage process of water injection in coal seam. Moreover, when the displacement pressure difference reaches 0.03 (lattice unit), the velocity ratio of CDEA-modified coal sample exceeds 100%; this means that when the displacement pressure difference surpasses 15.6 MPa, the water injection effect of CDEA-modified coal sample begins to be improved. These research findings offer a theoretical basis for enhancing water injection technology in coal mines.

DFT based analysis of pressure driven mechanical, opto-electronic, and thermoelectric properties in lead-free InGeX3 (X = Cl, Br) perovskites for solar energy applications

Lead halide perovskites have distinct physiochemical properties and demonstrate remarkable power conversion efficiency. We used density functional theory to investigate the electrical, optical, structural, and elastic features of non-toxic InGeCl3 and InGeBr3 halide perovskite compounds at different hydrostatic pressures, from 0 to 8 GPa. InGeCl3 and InGeBr3 halide perovskite exhibit noteworthy changes in their electronic and optical properties under different pressure conditions. When the pressure is 0 GPa, the direct bandgap for InGeCl3 is 0.886 eV, and for InGeBr3 it is 0.536 eV. This gap decreases as the pressure rises. Specifically, InGeBr3 exhibits conducting properties at 3 GPa due to its larger bromine atoms, whereas InGeCl3 requires a higher pressure of 6 GPa to achieve similar conductivity. This type of nature suggests that larger halogen atoms reduce the bandgap more effectively under pressure. As the pressure increases, the behavior of the lattice constant and unit cell volume decreases constantly, from 5.257 and 145.267 Å3 for InGeCl3 to 5.509 and 167.168 Å3 for InGeBr3 at 0 GPa for both compounds. When subjected to pressure, the bonds between In-X and Ge-X atoms experience compression, leading to a decrease in surface area and an enhancement in mechanical strength. Overall, the compounds exhibit characteristics of semiconductors, as evidenced by evaluations of their electrical properties. As pressure increases, the bandgap decreases linearly, narrowing until it aligns with the Fermi level, leading to a transition toward a metallic state. In addition, the pressure induces a rise in the electrical density of states around the Fermi level by displacing valence band electrons in an upward direction. As pressure increases, the electron density peak shifts to lower photon energy values. Notably, InGeCl3 exhibits a more pronounced shift in this peak compared to InGeBr3, indicating greater sensitivity to pressure. In terms of optical properties, both compounds demonstrate significant absorption coefficients in the visible region, suggesting their potential suitability for photovoltaic applications. The dielectric constant, absorption, and reflectivity values all increase gradually as pressure increases. The absorption spectra shift toward longer wavelengths. Furthermore, the mechanical properties analysis reveals that all InGeX3 compounds are mechanically stable up to 8 GPa pressure.

The investigation on the bending fatigue properties of Ti-6Al-4V lattice sandwich structures fabricated EB-PBF

In this study, lattice sandwich structures of Ti-6Al-4V alloy with simple cubic (SC) and rhombic dodecahedron (RD) unit cells were prepared by electron beam powder bed fusion (EB-PBF) technique. Through a combination of finite element simulation and experimental investigation, the effect of lattice cell shape and heat treatment on the bending fatigue performance of these lattice sandwich structures was studied. The results show that the underlying fatigue mechanism of the RD lattice sandwich structure is primarily dominated by cyclic ratcheting of the lattice. In contrast, for the SC lattice sandwich structure, the fatigue mechanism involves an interaction of cyclic ratcheting and fatigue crack initiation and propagation of the lattice. During the cyclic deformation process, the struts of the lattice unit cells in the SC lattice sandwich structure mainly undergo buckling deformation. Under the same bending deformation strain, the struts experience much higher stress in the SC lattice sandwich structure, making them more susceptible to crack initiation and resulting in a quick fatigue failure. Therefore, the fatigue performance of the RD lattice sandwich structure is significantly superior to that of the SC lattice sandwich structure. Heat treatment result in the enhanced plasticity of parent materials of the lattice sandwich structure, leading to improved resistance to cyclic strain accumulation during fatigue processes and an increase in fatigue strength.

Parameter-Independent Deformation Behaviour of Diagonally Reinforced Doubly Re-Entrant Honeycomb.

In this study, a novel unit cell design is proposed, which eliminates the buckling tendency of the auxetic honeycomb. The novel unit cell design is a more balanced, diagonally reinforced doubly re-entrant auxetic honeycomb structure (x-reinforced auxetic honeycomb for short). We investigated and compared this novel unit cell design against a wide parameter range. Compression tests were carried out on specimens 3D-printed with a special, unique, flexible but tough resin mixture. The results showed that the additional, centrally pronounced reinforcements resulted in increased deformation stability; parameter-independent, non-buckling deformation behaviour is achieved; however, the novel structure is no longer auxetic. Mechanical properties, such as compression resistance and energy absorption capability, also increased significantly-An almost four times increase can be observed. In contrast to the deformation behaviour (which became predictable and constant), the mechanical properties can be precisely adjusted for the desired application. This novel structure was also investigated in a highly accurate, validated finite element environment, which showed that critical stress values are formed in well-supported regions, meaning that critical failure is unlikely. Our novel lattice unit cell design elevated the auxetic honeycomb to the realm of modern, high performance and widely applicable lattice structures.

Fin angles optimization of water-cooled plate-fin heat sink based on anisotropic Darcy–Forchheimer theory

Flat heat sinks are devices for efficient heat dissipation and are important components in manufacturing. For example, in precision glass molding, it is utilized as a cooling plate contributing to improve the quality of the product by reducing residual stress and shape deviation, during the gradual cooling phase. In this study, we attempted to achieve precise temperature design of the cooling plate using variable lattice optimization. Fins allow for controlling the flow direction and achieving efficient heat transfer. Therefore, elliptical cylindrical fins were adopted as lattice unit cells, increasing the design freedom of the flow field. To account for the asymmetric flow field within the unit cell, we introduced an asymmetric tensor for the coefficients of the effective physical properties in the Darcy–Forchheimer equation. The proposed method's effectiveness and validity are discussed through numerical calculations with effective material properties, reproduced detailed shapes, and experimental verification. The optimization aiming to minimize the average temperature yielded a numerical simulation temperature decrease of 24.6 % and an experimental temperature decrease of 14.1 % compared to the benchmark model.

Helmets with lattice liners can mitigate traumatic brain injury from impacts

This study explores the effectiveness of architected lattice structures, specifically made of polyamide 12 (PA12) material, as potential helmet liners to mitigate traumatic brain injuries (TBI), with a focus on rotational acceleration. Evaluating three lattice unit cell topologies (simple cubic, dode-medium, and rhombic dodecahedron), the research builds upon prior investigations indicating that PA12 lattice liners may outperform conventional EPS liners. Employing a high-fidelity finite element male head model and utilizing direct and oblique impact scenarios, mechanical quantities, such as maximum principal strain (MPS) and shear strain, cumulative strain damage measure and intracranial pressure were measured at the tissue level in different brain regions. Results indicate that lattice liners, especially with dode-medium topology, exhibit promising reductions in brain tissue strains. On average, during oblique impacts, less than 1 % of the brain volume experienced an MPS level of 0.4 when the lattice liners were adopted, whereas that percentage was above 70 % with the expandable polystyrene (EPS) foam liners. Pressure-based assessments suggest that lattice liners may outperform EPS liners in oblique impacts, showcasing the limitations of EPS for effective TBI mitigation. Despite certain model limitations, this study emphasizes the need for advancements in helmet technology, particularly in the development of commercial lattice liners using additive manufacturing, to address the limitations of existing EPS liners in preventing rotational consequences of impacts and reducing TBI.

Double-foci acoustic metamaterial Luneburg lens

In this study, an acoustic Luneburg lens is proposed to achieve double foci. High energy density between the two foci can be obtained, which enables the realization of an ultra long acoustic jet between the two foci. The acoustic Luneburg lens is designed based on the gradient index acoustic metamaterial with lattice unit cells. Analytical and numerical analyses were performed to investigate the performance of double foci Luneburg lens. Results show that the double foci can be realized with acoustic jet length goes up to 30 times compared with the wavelength, which has multiple potential applications in medical imaging and energy harvesting.

Viability and antibacterial properties of Ba2-x AgxFeMoO6(x=0.0, 0.05) double perovskite oxides

Double perovskite Ba2FeMoO6 (BFMO) and doped with Ag ions were synthesized via the sol-gel method. The structural and antibacterial properties and viability of the samples were systematically investigated. The results of the Rietveld refinement indicated that both samples exhibited a single phase with cubic structure and space group Fm-3m. The substitution of Ag ions led to an increase in the unit cell and lattice parameters of BFMO, resulting in a change in the sample's morphology. The direct band gap (∼2eV) was measured for samples by Diffuse Reflectance Spectroscopy (DRS). Antibacterial activity of the Ba2FeMoO6 and Ba1.95Ag0.05FeMoO6 samples was assessed on the Gram-positive bacteria (Staphylococcus aureus) and the Gram-negative bacteria (Escherichia coli) using the spread plate method. The results showed that the samples were effective against Staphylococcus aureus but not against Escherichia coli. The sample doped with Ag ions showed a higher antibacterial effect than the pristine sample. An MTT assay of the normal fibroblast cell line treated with an Ag-doped sample showed a higher percentage of viability.

Comparison Study Between Inherited and Biogenic Calcium Carbonate Formation on The Surface Roots of Eucalyptus Trees Using X-Ray Technique and Field Observations

Background: This study aims to explain the mechanism of formation of calcite and it's to shed light on the synthetic carbonate crystallization on the surface of roots due to this mechanism is poorly understood and highly controversial. It is accepted that calcite formed originally from parent rocks and not in soils. Method: The origin of calcite in calcareous soil, in Aldiwaniya city, Iraq was investigated, by selecting two soil samples, one from the contact zone with root surface and the other sample from bulk soil and out of the root system. Results: X-ray diffraction showed that the calcite has two types, first one is well crystallized and ordered, with only minor deviations in lattice unit cell parameters relative to typical calcite. The clay-sized calcite (active calcite) is enriched in A and C-horizons relative to biogenic calcite and this mineral exist in larger size separates, and it's clearly appears in the shape of white crystals around eucalyptus root surfaces as shown in graphs. The results of X-ray showed that the appearance of smectite mineral which was diagnosed by the peak 14.21 A° in air dry magnesium-saturated treatment which expanded to 16.28 A° when saturated with ethylene glycol, where clarity of peak was increased, Also, the results showed that the appearance of peak 14.21 A° in magnesium saturation air dried, as a shoulder on the peak of the mica (10.14 A°). Conclusion: In general, calcite minerals have characteristic x-ray diffraction peaks at the following angles: 29.5, 31.7, 34.5, 35.7, 47.3, 56.4, and 63.4 degrees.

Impact of lattice structure on compressive performance and strength to weight ratio in continuous stereolithography

Lattice structures have emerged as a creative solution in additive manufacturing to achieve lightweight structures. Lattices offer the capability to reduce material usage, production costs, and weight while maintaining the structural integrity of a part. Due to its rise in popularity and excellent properties, lattice structures have been used in many applications, including but not limited to aerospace, bioengineering, and acoustics. This study aimed to explore the performance of different types of lattices on 3D additive manufactured parts by varying the type of unit cell. The three unit cells that were compared are body-centered cubic (BCC), face-centered cubic (FCC), and Fluorite. The lattice’s strength-to-weight ratio of different lattice unit cells when subjected to compression forces was evaluated. BCC and FCC have had considerable research while Fluorite has had far less research. This study aims to contribute new knowledge on the less studied Fluorite unit cell. The selected lattice structures were printed using continuous stereolithography (CSLA). This 3D printing process utilizes a photopolymerization reaction to solidify liquid resin layer by layer. Through this study, Fluorite demonstrated the highest strength-to-weight ratio among the three lattice structures examined, averaging 19,377 Nm/kg over five samples. Its potential for applications requiring high strength requirements is highlighted by this finding. Alternatively, BCC lattice structures had the weakest strength-to-weight ratio but exhibited remarkable structural resilience to deformation. The BCC lattice structures showed a more significant deformation before failure. Fluorite lattice structures are more suitable for high-strength applications, while BCC lattice structures can be used where high strength is not a primary parameter. This research aimed to determine the optimal lattice unit cell to implement in future choices in additive manufacturing applications.

Impact of Hydrogen Cage Occupancy on the Mechanical Properties and Elastic Anisotropies of sII Hydrates

In this study, we investigate the composition-dependent mechanical properties and elastic anisotropies of sII hydrogen hydrates using first-principles method. The evaluation of the elastic moduli and their direction dependency is achieved by computing the second-order elastic constants (SOECs) of the unit lattice. The various trends of elastic constants with the hydrogen composition of the cages introduces variations in the bonding strength, compressibility, stiffness, and shear properties of the structure which are captured by the Poisson's ratio, bulk, Young, and shear moduli, respectively. Elastic properties were found to be significantly influenced by the system's anisotropy, arising from the geometry of the cages and their unique arrangement within the lattice being affected by the increase in the hydrogen occupancy of the cages. The detailed analysis of elastic anisotropies revealed shifts in the strongest and weakest directions of the material with varying the hydrogen content of the cages. The Poisson's ratio captures the anisotropic bonding strengths within the crystal structure with the hydrogen composition of the lattice, explaining the reason behind the existence of strongest and weakest directions in terms of compression, tension, and shear forces. Taken together the established structure-property-composition relations will be useful in the design and optimization of hydrogen sII hydrates for energy storage applications.

High-pressure synthesis, crystal structure and anisotropic thermal/compressive behaviors of pseudo-ternary (V,Cr,Mo)P4

Pseudo-ternary CrP4-type (V,Cr,Mo)P4 was successfully synthesized at the conditions of 4 GPa and 900 °C by using a large volume multi-anvil-type press. SEM-EDS and STEM analyses revealed that the synthesized pseudo-ternary (V,Cr,Mo)P4 contains almost equimolar amounts of transition metals. The lattice parameters and unit cell volume of (V,Cr,Mo)P4 yield the average value of its end-members. The low-temperature (133 K–333 K) in-situ synchrotron XRD measurements demonstrated that the c-axis showed the highest coefficient of thermal expansion, followed by the b-axis and the a-axis. These thermal behaviors are the same as the binary end-members and pseudo-binary (V,Cr)P4. While from the high-pressure (P < 10 GPa) in-situ synchrotron XRD measurements, it was found that (V,Cr,Mo)P4 showed no phase transition and compression behaviors in which the c-axis was more compressible than the other two axes (b and a) and the b-axis was slightly compressible more than the a-axis. The zero-pressure bulk modulus of 106 (1) GPa and 122.1 (7) GPa were obtained for (V,Cr)P4 and (V,Cr,Mo)P4, respectively. MoP6 octahedral is highly distorted in comparison with VP4 and CrP4, thus the incorporation of MoP4 greatly yields incompressible behaviors of CrP4-type phosphides with respect to temperature and pressure.

Effect of the chemical substitution on structural parameters and microwave properties of the Co–Ni–Zn spinels

Co0.3Ni0.7-xZn xFe2O4 (where x = 0.00–0.7 with step 0.1) spinels or CNZF with constant Co concentration and varying of the Ni/Zn concentration were obtained by the ceramic method from initial oxides. Based on X-ray diffraction data it was concluded that all samples are single-phase (SG: Fd-3m). Concentration dependences of the lattice constant and unit cell volume correlate well with Vegard's rule and the average value of the Zn2+ and Ni2+ ionic radii. The microwave permeability was measured in the frequency range of 0.1–20 GHz by a coaxial technique. Two resonances on frequency dependences were observed. Analysis of the microwave properties and magnetostatic data showed that low-frequency magnetic peak was caused by a resonance of domain boundaries; high-frequency peak was due to a natural ferrimagnetic resonance. Obtained results opens broad perspectives for practical application of such kind of materials and possibility for microwave parameters control by the external magnetic field.