Magnetic Materials

Magnetic materials are the engineered alloys and ceramics that either store a magnetic field (permanent magnets) or carry and shape one (soft magnetic cores). They sit at the heart of every electric motor, generator, transformer, sensor, loudspeaker, and power converter, which makes them one of the most strategically important categories in industrial materials. The field splits sharply into two families: hard magnetic materials such as NdFeB, samarium cobalt, ferrite, and alnico that resist demagnetization, and soft magnetic materials such as electrical steel, soft ferrite, amorphous and nanocrystalline ribbon that magnetize and demagnetize almost freely.

Selecting a magnetic material is never a single-number decision. A permanent magnet is described by its remanence, coercivity, and energy product together with its temperature behavior and corrosion resistance, while a soft core is described by saturation, permeability, and core loss at a stated frequency. This guide decodes those parameters, the grade systems, and the governing standards so a procurement or design engineer can specify the right grade with confidence.

Large nickel-plated rectangular sintered neodymium-iron-boron (NdFeB) permanent magnet block held in an open hand for scale

Photo: Saimon (Researcher), CC BY-SA 4.0, via Wikimedia Commons

This guide is written for industrial purchasing engineers and design engineers. It covers 6 chapters from the hard versus soft split, through permanent-magnet grades, soft-core technologies, governing standards, and spec-sheet decoding, to selection decisions, with 7 selection FAQs and verified manufacturer references. All parameters reference the IEC 60404 series (60404-5, 60404-8-1, 60404-8-4, 60404-8-7), ASTM A977 and ASTM A1070, and published manufacturer datasheets.

Chapter 1 / 06

What Magnetic Materials Are

A magnetic material is any substance whose response to an applied magnetic field is strong enough to be exploited in engineering. The vast majority of practical magnetic materials are ferromagnetic or ferrimagnetic, meaning their atomic magnetic moments align cooperatively into domains that can be aligned by an external field and, in the case of permanent magnets, stay aligned after the field is removed. Iron, cobalt, and nickel are the three ferromagnetic elements at room temperature, and almost every commercial magnetic alloy is built around iron or cobalt with rare-earth, silicon, or oxide additions that tune the behavior.

The single most important diagram in the field is the hysteresis loop, the plot of magnetic flux density B against applied field strength H as the field is cycled. The shape of that loop defines whether a material is useful as a magnet or as a core. A tall, wide loop with high remanence and high coercivity describes a permanent magnet that stores energy. A tall, thin loop that encloses almost no area describes a soft material that carries flux with minimal loss. The enclosed loop area equals the energy dissipated per cycle, which is why soft cores chase the thinnest possible loop while permanent magnets chase the widest.

Three parameters read directly off the loop and recur throughout this guide. Remanence, written Br and measured in tesla (T) or millitesla (mT), is the flux density that remains when the applied field returns to zero. Coercivity, written Hc and measured in kiloamperes per meter (kA/m) or kilooersteds (kOe), is the reverse field needed to drive flux density back to zero, while intrinsic coercivity Hcj is the larger field needed to drive the material's own magnetization to zero. Saturation flux density Bs is the maximum flux the material can carry before further field produces almost no additional flux.

The industrial history is one of escalating energy density. Carbon steel magnets of the nineteenth century stored under 1 MGOe. Alnico, developed in the 1930s, reached 5 to 9 MGOe and dominated until the 1960s. Hard ferrite, introduced commercially by Philips in the 1950s, traded strength for cost and corrosion resistance. The rare-earth era began with samarium cobalt around 1970 and was transformed in 1983 when Sumitomo (Japan) and General Motors (USA) independently announced neodymium-iron-boron, which now reaches about 52 MGOe and powers most modern motors, hard drives, and wind turbines.

In parallel, soft magnetic materials advanced from plain iron to silicon-alloyed electrical steel (raising resistivity to cut eddy-current loss), then to rapidly solidified amorphous metal in the 1970s and nanocrystalline alloys in the late 1980s. Today the two families together represent a multi-billion-dollar market, with permanent magnets concentrated in rare-earth supply chains and soft materials concentrated in electrical steel and ferrite. The rest of this guide treats them as the two distinct product categories that buyers actually specify.

Chapter 2 / 06

Hard vs Soft Classification

The first and most consequential decision in any magnetic specification is whether the application needs a hard or a soft material. The two are not interchangeable, and the classification is set by coercivity. A common engineering threshold places soft materials below about 1 kA/m of coercivity and hard materials above it, though the practical gap is enormous: a transformer steel may have a coercivity of tens of amperes per meter, while a sintered NdFeB magnet exceeds 1,000 kA/m of intrinsic coercivity, a separation of four orders of magnitude.

PropertyHard (permanent) magnetsSoft magnetic materials
CoercivityHigh, >10 kA/m to >1,000 kA/mLow, often <1 kA/m
PermeabilityLowHigh, up to >100,000
Hysteresis loopWide, large areaNarrow, small area
FunctionStore and supply a fieldCarry and concentrate flux
Key specsBr, Hcj, BHmaxBs, permeability, core loss
Typical useMotors, sensors, speakersTransformer/motor cores, chokes

Hard magnetic materials, also called permanent magnets, retain a large magnetization after the magnetizing field is removed and act as standalone field sources. Their value lies in the maximum energy product BHmax, the largest product of B and H along the second quadrant of the demagnetization curve, expressed in kilojoules per cubic meter (kJ/m3) or mega-gauss-oersteds (MGOe, where 1 MGOe is about 7.96 kJ/m3). A higher BHmax means more field energy can be stored in a smaller volume, which is why rare-earth magnets enabled the miniaturization of motors and hard drives.

Soft magnetic materials are easy to magnetize and easy to demagnetize. They have high permeability so a small applied field produces a large flux, and they have a narrow loop so they waste little energy each time the field reverses. This makes them the conduit for alternating flux in transformers, motor and generator laminations, inductors, chokes, and electromagnet cores. Their figures of merit are saturation flux density Bs, initial and maximum permeability, and core loss in watts per kilogram at a stated frequency and flux density.

A useful mental model: the permanent magnet is the battery of magnetism, holding energy until needed, while the soft core is the wire of magnetism, conducting flux from place to place with minimal loss. Designs frequently use both. A brushless DC motor pairs NdFeB rotor magnets (hard) with a laminated electrical-steel stator (soft); a current sensor pairs a small bias magnet with a nanocrystalline core. Confusing the two roles is the most common conceptual error among engineers new to the category.

Within each family there are further subdivisions. Hard materials split into rare-earth (NdFeB, SmCo), ceramic (hard ferrite), and metallic-alloy (alnico) classes. Soft materials split into bulk metallic (electrical steel, FeNi permalloy, FeCo), ceramic (soft ferrite), amorphous, and nanocrystalline classes, plus soft magnetic composites (iron powder bonded in insulation) for three-dimensional flux paths. Chapters 3 and 4 take the hard and soft families in turn.

Chapter 3 / 06

Permanent Magnet Grades

Four permanent-magnet families cover essentially all industrial demand: sintered neodymium-iron-boron (NdFeB), samarium cobalt (SmCo), hard ferrite (ceramic), and alnico. Each occupies a distinct point on the price, strength, and temperature map, and no single family is best for every duty. The table below compares typical room-temperature properties; exact values depend on grade and supplier and should be confirmed against a current datasheet.

FamilyBHmax (MGOe)Remanence Br (mT)Max operating temp.Notes
Sintered NdFeB35 to 521,170 to 1,48080 to 230°C (by grade)Strongest; corrodes, needs coating
Samarium cobalt (SmCo)16 to 32850 to 1,150250 to 350°C (to 500+)Best high-temp stability; brittle
Hard ferrite (Y30)3 to 4370 to 400around 250°CCheapest; corrosion-resistant
Alnico5 to 9700 to 1,300up to 500°CTiny temp coefficient; low Hcj

Sintered NdFeB is the strongest commercial magnet. The numeric grade equals the nominal BHmax in MGOe, so N35 is about 35 MGOe (roughly 263 kJ/m3, Br near 1,170 mT) and N52 is about 52 MGOe (roughly 414 kJ/m3). The optional letter suffix encodes intrinsic coercivity and thus the maximum operating temperature: no suffix to about 80 degrees Celsius, M to 100, H to 120, SH to 150, UH to 180, EH to 200, and AH to about 230. Higher temperature grades raise Hcj at the cost of some Br. NdFeB is fundamentally rust-prone and is always plated with nickel-copper-nickel, zinc, or epoxy.

Samarium cobalt comes in two phases. SmCo5 (1:5) offers roughly 16 to 25 MGOe with service to about 250 degrees Celsius, while Sm2Co17 (2:17) offers roughly 18 to 32 MGOe with continuous service to about 350 degrees and special grades beyond 500. Its decisive advantage is a very small reversible temperature coefficient of remanence, around minus 0.03 to minus 0.04 percent per degree Celsius, roughly a third of NdFeB. SmCo also resists corrosion without plating. The trade-offs are higher cost and brittleness, so it is reserved for aerospace, downhole, sensor, and high-speed motor duties.

Hard ferrite, also called ceramic magnet, is made from iron oxide with barium or strontium carbonate. It is by far the cheapest permanent magnet and is fully corrosion resistant, which is why it dominates loudspeakers, simple DC motors, holding magnets, and magnetic separators by volume. The penalty is low strength: grade Y30 delivers a remanence of 370 to 400 mT, intrinsic coercivity of 180 to 220 kA/m, and a maximum energy product of only 26 to 30 kJ/m3 (about 3.3 to 3.8 MGOe). Ferrite also has a relatively large negative temperature coefficient of remanence, about minus 0.2 percent per degree.

Alnico, an aluminum-nickel-cobalt-iron alloy, was the dominant high-performance magnet from the 1930s to the 1960s and remains valuable for one reason: an exceptionally small temperature coefficient of remanence near minus 0.02 percent per degree, the lowest of any common magnet, plus a high Curie temperature allowing operation to about 500 degrees Celsius. Its weakness is very low coercivity, which makes it easy to demagnetize accidentally, so it survives mainly in precision sensors, instruments, magnetrons, and pickups where thermal stability beats raw strength. Cast and sintered alnico are specified by ASTM A1070.

Chapter 4 / 06

Soft Cores and Standards

Soft magnetic materials carry alternating flux, and the goal is always high saturation, high permeability, and low core loss at the operating frequency. The right choice is set almost entirely by frequency and flux density. Below a few kilohertz at high flux, bulk metallic alloys win; above tens of kilohertz, ferrites and nanocrystalline ribbon take over. The table below compares the four dominant soft-core families on the parameters that decide the design.

MaterialSaturation Bs (T)Frequency rangeRelative lossTypical use
Grain-oriented silicon steelabout 2.050 to 400 HzHigh at HFPower and distribution transformers
Amorphous (Fe-based, Metglas)1.5 to 1.6DC to ~20 kHzLow at LFEnergy-efficient transformer cores
Nanocrystalline (VITROPERM)1.2 to 1.251 kHz to ~1 MHzVery lowHF transformers, CM chokes, CTs
MnZn soft ferrite0.4 to 0.5to ~1 to 2 MHzLow at HFSMPS power transformers, chokes

Electrical (silicon) steel is iron alloyed with up to about 3.5 percent silicon to raise resistivity and cut eddy-current loss, supplied as thin laminations. Grain-oriented electrical steel (GOES) has its grains aligned along the rolling direction for very low loss and high permeability in that direction, reaching saturation around 2.0 T, and is the standard for transformer cores. Non-oriented electrical steel (NOES) has isotropic properties for rotating machines such as motors and generators. POSCO, Nippon Steel, and ArcelorMittal are leading producers. GOES is graded under IEC 60404-8-7 and NOES under IEC 60404-8-4.

Amorphous metal (metallic glass) is produced by rapidly quenching molten alloy into a thin ribbon so fast that no crystal structure forms. Iron-based amorphous (Metglas 2605 family) has saturation near 1.5 to 1.6 T and dramatically lower hysteresis loss than grain-oriented steel, cutting distribution-transformer no-load loss by roughly 70 percent, which is why it is widely used in energy-efficient distribution transformers. The trade-offs are lower saturation than silicon steel and a brittle ribbon that complicates core building.

Nanocrystalline alloy is amorphous ribbon annealed to nucleate iron grains about 10 nanometers across embedded in an amorphous matrix, combining the high permeability of amorphous metal with low loss approaching ferrite. Saturation is about 1.2 to 1.25 T, well above ferrite's 0.4 to 0.5 T, with permeability exceeding 20,000 at 100 kHz. It is the material of choice for high-frequency transformers, common-mode chokes, and precision current transformers. VACUUMSCHMELZE (VITROPERM) and Hitachi Metals (FINEMET) are the principal suppliers.

Soft ferrite is a sintered ceramic of iron oxide with manganese-zinc or nickel-zinc. Its high electrical resistivity nearly eliminates eddy-current loss at high frequency, which is why ferrite dominates switch-mode power supplies, RF, and EMI suppression despite a low saturation of 0.4 to 0.5 T. MnZn grades such as TDK N87 (initial permeability about 2,200, saturation about 490 mT at 25 degrees Celsius) and Ferroxcube 3C95 serve power conversion below about 1 to 2 MHz; NiZn grades serve signal and EMI duties above 1 MHz. The IEC 60404 series, with core loss measured by Epstein frame (IEC 60404-2) or single-sheet tester (IEC 60404-3), governs soft-material testing.

Chapter 5 / 06

Key Specification Parameters

A magnetic-material datasheet is meaningless without its measurement conditions, and reading it correctly is the core skill of specification. Permanent magnets and soft cores use different parameter sets, so this chapter decodes both. For permanent magnets the four numbers that drive every decision are remanence, intrinsic coercivity, maximum energy product, and the temperature coefficients.

Remanence (Br) is the flux density remaining at zero applied field, the practical measure of a magnet's strength, expressed in mT or T. Coercivity (Hcb) is the reverse field that drives flux density B to zero in a closed circuit, while intrinsic coercivity (Hcj) is the larger field that drives the magnet's own magnetization to zero. Hcj is the resistance to irreversible demagnetization and is the parameter that high-temperature grades maximize. A magnet with high Br but low Hcj is strong but fragile against opposing fields and heat.

Maximum energy product (BHmax) is the largest product of B and H on the demagnetization curve, in kJ/m3 or MGOe, and represents the maximum magnetic energy per unit volume the material can supply. It is the single figure most used to rank magnet families, but it describes only room-temperature strength, not thermal capability. The demagnetization curve also has a knee: if the operating point falls below the knee, the magnet suffers permanent loss, so the curve must be evaluated at the maximum operating temperature, not just at 25 degrees Celsius.

Temperature coefficients come in two forms. The reversible coefficient of remanence (alpha) gives the recoverable percent change in Br per degree Celsius: about minus 0.12 for NdFeB, minus 0.03 to minus 0.04 for SmCo, minus 0.2 for ferrite, and minus 0.02 for alnico. The reversible coefficient of intrinsic coercivity (beta) gives the change in Hcj per degree, about minus 0.4 to minus 0.6 for NdFeB. Beta is the hazard: as Hcj collapses with rising temperature the operating point can cross the knee and lose magnetization permanently.

Soft magnetic materials use a different set of parameters, summarized below:

  • Saturation flux density (Bs): the maximum flux the material can carry, in tesla. Higher Bs allows a smaller core; silicon steel reaches about 2.0 T, nanocrystalline about 1.2 T, ferrite only 0.4 to 0.5 T.
  • Permeability (mu): the ratio of flux density to applied field, dimensionless and relative to vacuum. Initial permeability matters at low signal; high values, above 100,000 for some nanocrystalline alloys, mean a small field drives large flux.
  • Core loss: energy dissipated per cycle, in watts per kilogram, the sum of hysteresis and eddy-current loss. It must be quoted at a specific frequency and flux density (for example, 1.5 T at 50 Hz) to be comparable.
  • Curie temperature: the temperature at which the material loses ferromagnetism entirely. It bounds the usable range and varies from about 215 degrees Celsius for some MnZn ferrites to over 700 for iron-based alloys.
  • Resistivity: higher resistivity cuts eddy-current loss; this is why silicon is alloyed into steel and why ferrites (insulators) excel at high frequency.

The recurring lesson is that every magnetic number is conditional. A remanence figure is meaningless without its test temperature; a core loss figure is meaningless without its frequency and flux density; a coercivity figure must specify whether it is Hcb or Hcj. When comparing quotes, demand that each value cite its measurement standard and conditions, or the comparison is invalid.

Chapter 6 / 06

Selection Decision Factors

Translating the preceding chapters into a specific grade follows an ordered sequence. Most selection errors come not from a single wrong value but from deciding strength before deciding function, or temperature, or environment. These eight steps work as a fixed RFQ template for either a permanent magnet or a soft core.

  1. Function first, hard or soft: Decide whether the part must supply a field (permanent magnet) or carry alternating flux (soft core). This single branch selects the entire material family and the parameter set you will specify.
  2. Operating temperature: Identify the worst-case temperature at the magnet or core, not the ambient. For permanent magnets this gates the family and grade suffix: NdFeB to 80 to 230 degrees by suffix, SmCo to 350 plus, alnico to 500. For soft cores, confirm the Curie temperature and loss rise with heat.
  3. Required strength or saturation: For magnets, set the minimum BHmax and Br for the magnetic circuit. For soft cores, set the required saturation Bs and operating flux density with margin so the core does not saturate at peak load.
  4. Frequency (soft cores) or demagnetizing circuit (magnets): Soft-core choice is dominated by frequency: silicon steel to a few hundred hertz, amorphous to tens of kilohertz, nanocrystalline and ferrite into the megahertz. For magnets, evaluate the load line against the demagnetization curve at peak temperature to keep the operating point above the knee.
  5. Corrosion and coating: NdFeB must be coated (NiCuNi, zinc, epoxy, or Parylene for medical and vacuum). SmCo, ferrite, and alnico need no plating. Soft cores in humid or chemical environments may need insulation or potting.
  6. Mechanical and dimensional: Rare-earth magnets and ferrites are brittle and cannot bear structural load; SmCo and ferrite chip easily. Confirm dimensional tolerances, machining limits, and whether the part is magnetized before or after assembly.
  7. Standards and certification: Cite the governing standard for every quoted value: IEC 60404-8-1 and ASTM A977 for permanent magnets, IEC 60404-8-4/8-7 for electrical steel, ASTM A1070 for cast and sintered alnico, plus any RoHS, REACH, or rare-earth-origin documentation the project requires.
  8. Total cost of ownership (TCO): Weigh material cost against performance per unit volume. Ferrite is cheapest per kilogram but bulky; NdFeB is costly per kilogram but tiny per joule. Rare-earth price volatility, supply security, and the cost of oversizing a cheaper grade all belong in the TCO model, not just the unit price.

One last commonly overlooked dimension is supply security and serviceability: rare-earth magnets depend on a concentrated supply chain, so confirm material origin, lead time, and whether a second source exists for the exact grade and suffix. For soft cores, verify that the supplier can hold core-loss tolerance batch to batch and provide measured loss curves at your operating point. Leading suppliers such as Shin-Etsu, Hitachi Metals, VACUUMSCHMELZE, Arnold Magnetic Technologies, TDK, Ferroxcube, POSCO, and Nippon Steel publish full datasheets and maintain application engineering, which de-risks long-production-life programs far more than a marginal price saving.

FAQ

What is the difference between hard and soft magnetic materials?

The dividing line is coercivity, the field needed to demagnetize the material. Hard (permanent) magnetic materials have high coercivity, typically above 10 kA/m and often above 1,000 kA/m for sintered NdFeB, so they retain magnetization after the applied field is removed and act as standalone field sources. Soft magnetic materials have low coercivity, often below 1 kA/m, plus high permeability, so they magnetize and demagnetize easily and are used to carry and concentrate flux in transformer cores, motor laminations, and inductors. Hard materials are graded by remanence Br, intrinsic coercivity Hcj, and maximum energy product BHmax. Soft materials are graded by saturation flux density Bs, permeability, and core loss in watts per kilogram. They are different product families with different test standards under the IEC 60404 series.

What does an NdFeB grade like N52 actually mean?

The number in an NdFeB grade is the nominal maximum energy product BHmax in MGOe (mega-gauss-oersteds). N52 means about 52 MGOe, which is roughly 414 kJ/m3 (multiply MGOe by about 7.96). N35 means about 35 MGOe or roughly 263 kJ/m3 with a remanence near 1,170 mT, while N42 sits near 1,280 mT. A higher number means a stronger magnet at room temperature but does not describe temperature capability. The optional letter suffix encodes intrinsic coercivity and therefore maximum operating temperature: no suffix is about 80 degrees Celsius, M about 100, H about 120, SH about 150, UH about 180, EH about 200, and AH up to roughly 230. So N52 is the strongest standard grade but only to 80 degrees, whereas N42SH trades some strength for service to 150 degrees.

Why choose SmCo over NdFeB when NdFeB is stronger?

NdFeB has the highest room-temperature energy product of any commercial magnet, up to about 52 MGOe, but it loses strength quickly with heat and corrodes without a coating. Samarium cobalt (SmCo) has a lower energy product, roughly 16 to 32 MGOe, but a far smaller reversible temperature coefficient of remanence, near minus 0.03 to minus 0.04 percent per degree Celsius versus minus 0.09 to minus 0.12 for NdFeB. Sm2Co17 grades operate continuously to about 350 degrees Celsius, with special grades to 500 degrees or higher, and resist corrosion without plating. Above roughly 150 to 180 degrees Celsius, SmCo delivers both higher usable energy product and better stability than NdFeB, so it is preferred in aerospace, downhole, high-speed motors, and sensor magnets where temperature and reliability outweigh raw strength.

How do I read the temperature coefficients on a permanent magnet datasheet?

Two coefficients matter. The reversible temperature coefficient of remanence, often written as alpha, gives the percent change in Br per degree Celsius and is recoverable on cooling: about minus 0.12 for NdFeB, minus 0.03 to minus 0.04 for SmCo, minus 0.2 for ferrite, and only minus 0.02 for alnico. The reversible coefficient of intrinsic coercivity, often written as beta, gives the percent change in Hcj per degree and is about minus 0.4 to minus 0.6 for NdFeB. Beta is the dangerous one: as Hcj falls with rising temperature, the magnet can cross its knee and suffer irreversible loss that does not recover on cooling. That is why high-temperature grades raise Hcj, not Br. Always confirm the maximum operating temperature for your worst-case demagnetizing circuit, not just the bench temperature.

What is the difference between MnZn and NiZn ferrite cores?

Both are soft ferrite ceramics for inductor and transformer cores, but they cover different frequency bands. Manganese-zinc (MnZn) ferrite has high initial permeability, roughly 1,400 to 15,000, and higher saturation flux density near 400 to 500 mT, with low loss below about 1 to 2 MHz, so it dominates power transformers and chokes such as TDK N87 or Ferroxcube 3C95. Nickel-zinc (NiZn) ferrite has lower permeability and lower saturation but much higher electrical resistivity, so it has low eddy-current loss above about 1 MHz and is used for EMI suppression, RF inductors, and antenna rods. As a rule of thumb, choose MnZn below about 1 MHz for power and NiZn above 1 MHz for signal and suppression. NiZn also tolerates higher temperatures and is mechanically harder.

When are amorphous or nanocrystalline cores worth the cost over silicon steel or ferrite?

Amorphous (metallic glass) and nanocrystalline ribbon cores fill the gap between silicon steel and ferrite. Nanocrystalline alloys such as VAC VITROPERM or Hitachi FINEMET reach high permeability, well over 20,000 at 100 kHz, with saturation around 1.2 to 1.25 T, far above ferrite's 0.4 to 0.5 T, and core loss far below silicon steel at high frequency. They are worth the premium in high-frequency power transformers, common-mode chokes, current transformers, and EV onboard chargers where size and efficiency dominate. Amorphous iron-based cores (Metglas 2605) cut distribution-transformer no-load loss by roughly 70 percent versus grain-oriented steel. Below a few kilohertz at high flux, grain-oriented silicon steel still wins on saturation and cost; above tens of kilohertz at low flux, ferrite can be cheaper. The decision is frequency, flux density, and loss budget.

Which standards govern magnetic material specifications and testing?

The IEC 60404 series is the backbone. IEC 60404-8-1 specifies minimum magnetic properties and dimensional tolerances for permanent (magnetically hard) materials, while IEC 60404-5 defines how to measure demagnetization curves and recoil lines for those magnets. For soft materials, IEC 60404-8-4 covers non-oriented electrical steel and IEC 60404-8-7 covers grain-oriented steel, with core loss measured by the Epstein frame (IEC 60404-2) or single-sheet tester (IEC 60404-3). In North America, ASTM A977 covers hysteresigraph testing of high-coercivity permanent magnets, and ASTM A1070 covers cast and sintered alnico. Chinese GB/T standards mirror many IEC grades. Always confirm that a quoted Br, Hcj, BHmax, or core loss cites the test standard, test temperature, and flux density, because numbers are meaningless without those conditions.

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